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# simdataset

simdataset simulates and-or contaminates a dataset given the parameters of a finite mixture model with Gaussian components

## Syntax

• X=simdataset(n, Pi, Mu, S)example
• X=simdataset(n, Pi, Mu, S,Name,Value)example
• [X,id]=simdataset(___)example

## Description

simdataset(n, Pi, Mu, S) generates a matrix of size $n$-by-$p$ containing $n$ observations $p$ dimensions from $k$ groups. More precisely, this function produces a dataset of n observations from a mixture model with parameters 'Pi' (mixing proportions), 'Mu' (mean vectors), and 'S' (covariance matrices). Mixture component sample sizes are produced as a realization from a multinomial distribution with probabilities given by the mixing proportions. For example, if n=200, k=4 and Pi=[0.25, 0.25, 0.25, 0.25] function Nk1=mnrnd( n-k, Pi) is used to generate k integers (whose sum is n-k) from the multinomial distribution with parameters n-k and Pi. The size of the groups is given by Nk1+1. The first Nk1(1)+1 observations are generated using centroid Mu(1,:) and covariance S(:,:,1), ..., the last Nk1(k)+1 observations are generated using centroid Mu(k,:) and covariance S(:,:,k).

DETAILS.

To make a dataset more challenging for clustering, a user might want to simulate noise variables or outliers. The optional parameter 'noiseunits' controls the number and the type of outliers which must be added. The optional parameter 'noisevars' controls the number and the type of noise variables which must be added (it is possible to control the distribution, the interval and the number). Finally, the user can apply an inverse Box-Cox transformation providing a vector of coefficients 'lambda'. The value 1 implies that no transformation is needed for the corresponding coordinate. It is also possible to add outliers to an existing dataset by simply suppling as first argument the matrix of existing data.

 X =simdataset(n, Pi, Mu, S) Example of mixture generation.

 X =simdataset(n, Pi, Mu, S, Name, Value) Generate 4 groups in 2 dimensions.

 [X, id] =simdataset(___) Generate 4 groups in 2 dimensions and add outliers from uniform distribution.

## Examples

expand all

### Example of mixture generation.

out = MixSim(4,2,'BarOmega',0.01);
n=60;
[X,id]=simdataset(n, out.Pi, out.Mu, out.S);
%  Simulate dataset with 10 outliers
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noiseunits',10);
%  Simulate dataset with 100 outliers
out = MixSim(4,3,'BarOmega',0.1);
n=300;
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noiseunits',100);
spmplot(X,id);

### Generate 4 groups in 2 dimensions.

rng('default')
rng(100)
out = MixSim(4,2,'BarOmega',0.01);
n=300;
[X,id]=simdataset(n, out.Pi, out.Mu, out.S);
spmplot(X,id);
title('4 groups without noise and outliers')

### Generate 4 groups in 2 dimensions and add outliers from uniform distribution.

rng('default')
rng(100)
out = MixSim(4,2,'BarOmega',0.01);
n=300;
noisevars=0;
noiseunits=3000;
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from uniform')

## Related Examples

expand all

### Add outliers generated from Chi2 with 5 degrees of freedom.

out = MixSim(4,2,'BarOmega',0.01);
n=300;
noisevars=0;
noiseunits=struct;
noiseunits.number=3000;
% Add asymmetric very concentrated noise
noiseunits.typeout={'Chisquare5'};
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from $\chi^2_5$','Interpreter','Latex')

### Add outliers generated from Chi2 with 40 degrees of freedom.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=0;
noiseunits=struct;
noiseunits.number=3000;
noiseunits.typeout={'Chisquare40'};
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from $\chi^2_{40}$','Interpreter','Latex')

### Add outliers generated from normal distribution.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=0;
noiseunits=struct;
noiseunits.number=3000;
noiseunits.typeout={'normal'};
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from normal distribution','Interpreter','Latex')

### Add outliers generated from Student T with 5 degrees of freedom.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=0;
noiseunits=struct;
noiseunits.number=3000;
noiseunits.typeout={'T5'};
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from Student T with 5 degrees if freedom','Interpreter','Latex')
Warning: it was not possible to generate 3000 outliers
in 30000 replicates in the interval [0.16444--1.0276]
Number of values which was possible to generate is equal to 30
Please modify the type of outliers using option 'typeout'
or increase input option 'alpha'
The value of alpha now is 0.001
Outliers have been generated according to T5
Warning: Output matrix X will have just 330 rows and not 3300

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars='';
noiseunits=struct;
noiseunits.number=3000;
noiseunits.typeout={'componentwise'};
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with component wise outliers','Interpreter','Latex')

### Add outliers generated from Chisquare and T distribution.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=0;
noiseunits=struct;
noiseunits.number=5000*ones(2,1);
noiseunits.typeout={'Chisquare3','T20'};
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from $\chi^2_{3}$ and $T_{20}$','Interpreter','Latex')

### Add outliers from Chisquare and T distribution and use a personalized value of alpha.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=0;
noiseunits=struct;
noiseunits.number=5000*ones(2,1);
noiseunits.typeout={'Chisquare3','T20'};
noiseunits.alpha=0.2;
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from $\chi^2_{3}$ and $T_{20}$ and $\alpha=0.2$','Interpreter','Latex')

### Add outliers from Chi2 and point mass contamination and add one noise variable.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=struct;
noisevars.number=1;
noiseunits=struct;
noiseunits.number=[100 100];
noiseunits.typeout={'pointmass' 'Chisquare5'};
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from $\chi^2_{5}$ and point mass $+1$ noise var','Interpreter','Latex')

### Example of the use of personalized interval to generate outliers.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noiseunits=struct;
noiseunits.number=1000;
noiseunits.typeout={'uniform'};
% Generate outliers in the interval [-1 1] for the first variable and
% interval [1 2] for the second variable
noiseunits.interval=[-1 1;
1 2];
% Finally add a noise variable
noisevars=struct;
noisevars.number=1;
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers from uniform using a personalized interval $+1$ noise var','Interpreter','Latex')

### Example of the use of personalized interval to generate outliers (1).

Generate 1000 outliers from uniform in the interval [-2 3] and 1000 units using componentwise contamination in the interval [-2 3]

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noiseunits=struct;
noiseunits.number=[1000 1000];
noiseunits.typeout={'uniform' 'componentwise'};
noiseunits.interval=[-2 3];
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups with outliers componentwise and from uniform in interval [-2 3]','Interpreter','Latex')

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=struct;
noisevars.number=[2 3];
noisevars.distribution={'Chisquare3','T20'};
noiseunits='';
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id,[],'box');
title('4 groups in 2 dims with 5 noise variables. First two from $\chi^2_{3}$ and last three from $T_{20}$','Interpreter','Latex')

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=struct;
noisevars.number=[1 2];
noisevars.distribution={'Chisquare3','T2'};
noiseunits='';
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups in 2 dims with 3 noise variables. First from $\chi^2_{3}$ and last two from $T_{2}$','Interpreter','Latex')

### Add 3 noise variables and use 'minmax' interval.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=struct;
noisevars.number=[1 2];
noisevars.distribution={'Chisquare3','T20'};
noisevars.interval='minmax';
noiseunits='';
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups in 2 dims with 3 noise variables with ''minimax'' interval','Interpreter','Latex')

### Add 3 noise variables and use a personalized interval for each variable.

n=300;
out = MixSim(4,2,'BarOmega',0.01);
noisevars=struct;
noisevars.number=[1 2];
noisevars.distribution={'Chisquare3','T20'};
noiseunits='';
% In this example we supply min and max for each noise variable
v1=sum(noisevars.number);
noisevars.interval=[3*ones(1,v1); 10*ones(1,v1)];
[X,id]=simdataset(n, out.Pi, out.Mu, out.S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(X,id);
title('4 groups in 2 dims with 3 noise variables with personalized interval','Interpreter','Latex')

### Add noise to an existing dataset.

Add outliers generated from uniform distribution to the IRIS dataset

load fisheriris;
Y=meas;
Mu=grpstats(Y,species);
S=zeros(4,4,3);
S(:,:,1)=cov(Y(1:50,:));
S(:,:,2)=cov(Y(51:100,:));
S(:,:,3)=cov(Y(101:150,:));
pigen=ones(3,1)/3;
% Add 100 outliers and specify a very small value of alpha
noisevars=0;
noiseunits=struct;
noiseunits.number=100;
noiseunits.alpha=0.000001;
% In this case the first argument which is supplied to simdataset is
% the original matrix X
[Ywithnoise,id]=simdataset(Y, pigen, Mu, S,'noisevars',noisevars,'noiseunits',noiseunits);
spmplot(Ywithnoise,id,[],'box');
title('4 groups with outliers from uniform')

## Input Arguments

### n — sample size or input matrix. Scalar.

Scalar or matrix of size n-by-v. If n is a scalar it is interpreted as the sample size of the dataset which must be simulated. On the other hand, if n is a n-by-v it is interpreted as a matrix of size n-by-v which has to be contaminated with optional input arguments 'noiseunits' and 'noisevars'.

Data Types: single| double

### Pi — Mixing proportions. Vector.

Vector of size k containing mixing proportions. The sum of the elements of Pi is 1.

Data Types: single| double

### Mu — centroids. Matrix.

Matrix of size k-by-v containing (in the rows) the centroids of the k groups.

Data Types: single| double

### S — Covariance matrices. 3D array.

3D array of size v-by-v-by-k containing covariance matrices of the k groups.

Data Types: single| double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as  Name1,Value1,...,NameN,ValueN.

Example:  'noiseunits', 10 , 'noisevars', 5 , 'lambda',[1 1 0]; , 'R_seed',1;

### noiseunits —number of type of outlying observations.scalar | structure.

Missing value, scalar or structure.

This input parameter specifies the number and type of outlying observations. The default value of noiseunits is 0.

- If noiseunits is a scalar t different from 0, then t units from the uniform distribution in the interval min(X) max(X) are generated in such a way that their squared Mahalanobis distance from the centroids of each existing group is larger then the quantile 1-0.999 of the Chi^2 distribution with p degrees of freedom. In order to generate these units the maximum number of attempts is equal to 10000.

- If noiseunits is a structure it may contain the following fields:

number = scalar, or vector of length f. The sum of the elements of vector 'number' is equal to the total number of outliers which are simulated.

alpha = scalar or vector of legth f containing the level(s) of simulated outliers. The default value of alpha is 0.001.

maxiter = maximum number of trials to simulate outliers.

The default value of maxiter is 10000.

interval= missing value or vector of length 2 or matrix of size 2-by-v which controls the min and max of the generated outliers for each dimension.

If interval is empty (default), the outliers are simulated in the interval min(X) max(X).

If interval is a vector of length(2), outliers for each variables are simulated in the range interval(1) and interval(2).

If interval is a 2-by-v matrix outliers are simulated in:

interval(1,1) interval (2,1) for variable 1 ...

interval(1,v) interval (2,v) for variable v typeout = list of length f containing the type of outliers which must be simulated. Possible values for typeout are:

* unif (or uniform), if the outliers must be generated using the uniform distribution;

* norm (or normal), if the outliers must be generated using the normal distribution;

* Chisquarez, if the outliers must be generated using the Chi2 distribution with z degrees of freedom;

* Tz or tz, if the outliers must be generated using the Student T distribution with z degrees of freedom;

* pointmass, if the outliers are concentrated on a particular point;

* componentwise, if the outliers must have the same coordinates of the existing rows of matrix X apart from a single coordinate (which will be the min or max in that particular dimension or the min or max specified in interval).

For example, the code:

noiseunits=struct;

Value Description
number

[100 100];

typeout

{'uniform' 'componentwise'};

interval

[-2 2];

adds 200 outliers, the first 100 generated using a uniform distribution and the last 100 using componentwise scheme. Outliers are generated in the interval [-2 2] for each variable.

Example:  'noiseunits', 10

Data Types: double

### noisevars —Type of noise variables.scalar | structure.

Empty value, scalar or structure.

- If noisevars is not specified or is an empty value (default) no noise variable is added to the matrix of simulated data.

- If noisevars is a scalar equal to r, then r new noise variables are added to the matrix of simulated data using the uniform distribution in the range [min(X) max(X)].

- If noisevars is a structure it may contain the following fields:

Value Description
number

a scalar or a vector of length f. The sum of elements of vector 'number' is equal to the total number of noise variables to be addded.

distribution

string or cell array of strings of length f which specifies the distribution to be used to simulate the noise variables.

If field distribution is not present then the uniform distribution is used to simulate the noise variables.

String 'distribution' can be one of the following values:

* uniform = uniform distribution * normal = normal distribution * t or T followed by a number which controls the degrees of freedom. For example, t6 specifies to generate the data according to a Student T with 6 degrees of freedom.

* chisquare followed by a number which controls the degreess of freedom. For example, chisquare8 specifies to generate the data according to a Chi square distribution with 8 degrees of freedom.

interval

string or vector of length 2 or matrix of size 2-by-f (where f is the number of noise variables) which controls for each element of vector 'number' or each element of cell 'distribution', the min and max of the noise variables. For example, interval(1,3) and interval(2,3) are respectively the minimum and maximum values of simulated the data for the third noise variable If interval is empty (default), the noise variables are simulated uniformly between the smallest and the largest coordinates of mean vectors.

If interval is 'minmax' the noise varaibles are simulated uniformly between the smallest and the largest coordinates of the simulated data matrix.

For example, the code:

noisevars=struct;

noisevars.number=[3 2];

noisevars.distribution={'Chisquare5' 'T3'};

noisevars.interval='minmax';

adds 5 noise variables, the first 3 generated using the Chi2 with 5 degrees of freedom and the last two using the Student t with 3 degrees of freedom. Noise variables are generated in the interval min(X) max(X).

Example:  'noisevars', 5

Data Types: double

### lambda —Transformation coefficients.vector.

Vector of length v containing inverse Box-Cox transformation coefficients. The value false (default) implies that no transformation is applied to any variable.

Example:  'lambda',[1 1 0];

Data Types: double

### R_seed —random numbers from R language.scalar.

Scalar > 0 for the seed to be used to generate random numbers in a R instance. This is used to check consistency of the results obtained with the R package MixSim. See file Connect_Matlab_with_R_HELP to know how to connect MATLAB with R. This option requires the installation of the R-(D)COM Interface. Default is 0, i.e. random numbers are generated by matlab.

Example:  'R_seed',1;

Data Types: double

## Output Arguments

### X —Simulated dataset.  Matrix

Simulated dataset of size (n + noiseunits)-by-(v + noisevars).

Noise coordinates are provided in the last noisevars columns.

### id —Classification vector.  Vector

Classification vector of length n + noiseunits. Negative numbers represents the groups associated to the contaminated units.

REMARK: If noiseunits outliers could not be generated a warning is produced. In this case matrix X and vector id will have less than n + noiseunits rows.

## References

Maitra, R. and Melnykov, V. (2010), Simulating data to study performance of finite mixture modeling and clustering algorithms, "The Journal of Computational and Graphical Statistics", Vol. 19, pp. 354-376. [to refer to this publication we will use "MM2010 JCGS"]

Melnykov, V., Chen, W.-C. and Maitra, R. (2012), MixSim: An R Package for Simulating Data to Study Performance of Clustering Algorithms, "Journal of Statistical Software", Vol. 51, pp. 1-25.

Davies, R. (1980), The distribution of a linear combination of chi-square random variables, "Applied Statistics", Vol. 29, pp. 323-333.

Riani, M., Cerioli, A., Perrotta, D. and Torti, F. (2015), Simulating mixtures of multivariate data with fixed cluster overlap in FSDA, "Advances in data analysis and classification", Vol. 9, pp. 461-481.

[DOI 10.1007/s11634-015-0223-9].


text: Finding quadratic function from two points and known max value - Printable Version +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: Not HP Calculators (/forum-7.html) +--- Forum: Not remotely HP Calculators (/forum-9.html) +--- Thread: Finding quadratic function from two points and known max value (/thread-8945.html) Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017 12:57 PM I need to work this into a C# program for generating points along an arc, and the math is making me cross-eyed. Essentially, I know two points on the parabola, I know the y value of the maximum (but not x), and I know that the second derivative is negative, i.e. the coefficient of x^2 is negative. All the typical approaches of treating it as a system of 3 linear equations seem to rely on having three known points, but I've got two, plus an additional equation 2*a*x_3+b=0 where x_3 is unknown. Is there a straight-forward way of tackling this without pulling a bunch of numeric and matrix libraries into the program? RE: Finding quadratic function from two points and known max value - grsbanks - 08-31-2017 01:03 PM Short answer: nope. Just like a linear function can be defined with 2 points, a quadratic function needs 3 points to be defined. The best you can get is a family of functions that pass through the two points you know and have the known maximum. In the more general case, a polynomial of order n (ax^n+bx^(n-1)+cx^(n-2)...) requires n+1 points and you end up solving a system of n+1 simultaneous equations to get your coefficients. RE: Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017 01:09 PM Well, I've got two points, plus a known maximum value. Intuitively, I think that uniquely defines a parabola, though I could be wrong. RE: Finding quadratic function from two points and known max value - Nigel (UK) - 08-31-2017 02:46 PM I think that there are two possible solutions, but I could be wrong too! Let the maximum y-value be $$c$$ and let this occur at $$x=b$$, where $$b$$ is as yet unknown. Then the parabola's equation can be written as $$y=c-a(x-b)^2,$$ where $$a$$ is another unknown. We know two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ on the curve. Substituting these into the equation above gives \eqalign{y_1=c-a(x_1-b)^2\cr y_2=c-a(x_2-b)^2\cr}. Rearranging gives $${x_1-b\over x_2-b}=\pm\sqrt{c-y_1\over c-y_2}.$$ The value of $$b$$ is between $$x_1$$ and $$x_2$$ if the right-hand side is negative and outside this range otherwise; this is where the ambiguity comes in. Since $$c$$, $$y_1$$, and $$y_2$$ are known the right hand side can be computed and a sign chosen; this means that $$b$$ can be found, and then either of the previous equations can be used to give $$a$$. Is this what you are looking for? Nigel (UK) RE: Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017 05:02 PM Oh yes, I see what you mean. In the cases I'm dealing with, the x value of the vertex will always lie between the two known points, so that narrows it down to a single possible function. RE: Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017 05:31 PM Yup, right on the money. Just find b with "b = (x2 * sqrt((c - y1)/(c - y2)) + x1) / (sqrt((c - y1)/(c - y2)) + 1)", then find a with "a = (c - y1)/(x1 - b)^2". Then you've got your a, b, and c for the vertex form, and you can calculate y values. Thanks Nigel.

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# 2015 - Sem 1 - Hauser - Rond  | enregistrements trouvés : 8

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## Post-edited  Interview at CIRM: Michael Artin Artin, Michael (Personne interviewée) | CIRM (Editeur )

Michael ARTIN participated in the "Artin Approximation and Infinite dimensional Geometry" event organized at CIRM in March 2015, which was part of the Jean-Morlet semester held by Herwig Hauser. Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry and also generally recognized as one of the outstanding professors in his field. Artin was born in Hamburg, Germany, and brought up in Indiana. His parents were Natalia Jasny (Natascha) and Emil Artin, a preeminent algebraist of the 20th century. In 2002, Artin won the American Mathematical Society's annual Steele Prize for Lifetime Achievement. In 2005, he was awarded the Harvard Centennial Medal. He won the Wolf Prize in Mathematics. He is also a member of the National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences, the American Association for the Advancement of Science, the Society for Industrial and Applied Mathematics, and the American Mathematical Society.
Michael ARTIN participated in the "Artin Approximation and Infinite dimensional Geometry" event organized at CIRM in March 2015, which was part of the Jean-Morlet semester held by Herwig Hauser. Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry and also generally recognized as one of the outstanding professors ...

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## Post-edited  Commutative algebra for Artin approximation - Part 1 Hauser, Herwig (Auteur de la Conférence) | CIRM (Editeur )

In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université
In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach ...

13J05

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## Multi angle  The algebra and model theory of transseries Aschenbrenner, Matthias (Auteur de la Conférence) | CIRM (Editeur )

The concept of a "transseries" is a natural extension of that of a Laurent series, allowing for exponential and logarithmic terms. Transseries were introduced in the 1980s by the analyst Écalle and also, independently, by the logicians Dahn and Göring. The germs of many naturally occurring real-valued functions of one variable have asymptotic expansions which are transseries. Since the late 1990s, van den Dries, van der Hoeven, and myself, have pursued a program to understand the algebraic and model-theoretic aspects of this intricate but fascinating mathematical object. A differential analogue of “henselianity" is central to this program. Last year we were able to make a significant step forward, and established a quantifier elimination theorem for the differential field of transseries in a natural language. My goal for this talk is to introduce transseries without prior knowledge of the subject, and to explain our recent work.
The concept of a "transseries" is a natural extension of that of a Laurent series, allowing for exponential and logarithmic terms. Transseries were introduced in the 1980s by the analyst Écalle and also, independently, by the logicians Dahn and Göring. The germs of many naturally occurring real-valued functions of one variable have asymptotic expansions which are transseries. Since the late 1990s, van den Dries, van der Hoeven, and myself, have ...

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## Multi angle  Tannaka duality and formal glueings Hall, Jack (Auteur de la Conférence) | CIRM (Editeur )

After recalling classical Tannaka duality for finite groups, I will discuss an extension to equivariant algebraic geometry, and more generally to algebraic stacks. Surprisingly, this is related to formal glueings and Neron-Popescu desingularization.

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## Multi angle  Stabilisation in algebraic geometry Draisma, Jan (Auteur de la Conférence) | CIRM (Editeur )

Across many mathematical disciplines one encounters projective systems of algebraic varieties indexed by a combinatorial datum, such as a natural number, a finite graph, or a lattice polytope. As the datum grows, the algebraic complexity of the corresponding variety (measured, for instance, in terms of its defining equations or higher-order syzygies) typically increases. But in good cases it eventually stabilises in a well-defined manner, especially when the family admits a direct system of sufficiently large symmetry groups. Exactly when this stabilisation phenomenon can be expected is still poorly understood, and this question motivates much current research activity in algebraic geometry and adjacent branches. After a brief general setup, which involves passing to the projective limit of the varieties and the direct limit of their symmetry groups, I will discuss a number of concrete instances where stabilisation occurs, both from classical algebraic geometry and from other areas of mathematics.
Across many mathematical disciplines one encounters projective systems of algebraic varieties indexed by a combinatorial datum, such as a natural number, a finite graph, or a lattice polytope. As the datum grows, the algebraic complexity of the corresponding variety (measured, for instance, in terms of its defining equations or higher-order syzygies) typically increases. But in good cases it eventually stabilises in a well-defined manner, ...

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## Multi angle  Pfaffian functions: real and non-archimedean, and an application to counting rational points Cluckers, Raf (Auteur de la Conférence) | CIRM (Editeur )

We will present some of the original definitions, results, and proof techniques about Pfaffian functions on the reals by Khovanskii.
A simple example of a Pfaffian function is an analytic function $f$ in one variable $x$ satisfying a differential equation $f^\prime = P(x,f)$ where $P$ is a polynomial in two variables. Khovanskii gives a notion of complexity of Pfaffian functions which in the example is just the degree of $P$. Using this complexity, he proves analogues of Bézout's theorem for Pfaffian curves (say, zero loci of Pfaffian functions in two variables), with explicit upper bounds in terms of the ocurring complexities.
We explain a recent application by J. Pila and others to a low-dimensional case of Wilkie's conjecture on rational points of bounded height on restricted Pfaffian curves. The result says that the number of rational points of height bounded by $T$, on a transcendental restricted Pfaffian curve, grows at most as a power of log$(T)$ as $T$ grows. This improves the typical upper bound $T^\epsilon$ in Pila-Wilkie's results in general o-minimal structures, the improvement being due to extra geometric Bézout-like control.
In the non-archimedean setting, I will explain analogues of some of these results and techniques, most of which are (emerging) work in progress with L. Lipshitz, F. Martin and A. Smeets. Some ideas in this case come from work by Denef and Lipshitz on variants of Artin approximation in the context of power series solution.
We will present some of the original definitions, results, and proof techniques about Pfaffian functions on the reals by Khovanskii.
A simple example of a Pfaffian function is an analytic function $f$ in one variable $x$ satisfying a differential equation $f^\prime = P(x,f)$ where $P$ is a polynomial in two variables. Khovanskii gives a notion of complexity of Pfaffian functions which in the example is just the degree of $P$. Using this ...

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## Multi angle  Commutative algebra for Artin approximation - Part 3 Hauser, Herwig (Auteur de la Conférence) | CIRM (Editeur )

In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université
In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach ...

13J05

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## Multi angle  Commutative algebra for Artin approximation - Part 2 Hauser, Herwig (Auteur de la Conférence) | CIRM (Editeur )

In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université
In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach ...

14B25

Z


text:
• 论文与研究报告 •

### 基于随机森林算法的纤维板表面缺陷识别

1. 1. 山东建筑大学信息与电气工程学院 济南 250101;
2. 中国林业科学研究院木材工业研究所 北京 100091
• 收稿日期:2018-04-02 修回日期:2018-08-30 出版日期:2018-11-25 发布日期:2018-12-04
• 基金资助:
中国林业科学研究院中央级公益性科研院所基本科研业务费专项资金（CAFYBB2018MB002）；泰山学者优势特色学科人才团队（2015162）。

### Surface Defect Recognition of Fibreboard Based on Random Forest

Liu Chuanze1, Wang Xiao2, Chen Longxian1, Guo Hui2, Luo Rui1, Zhou Yucheng1

1. 1. College of Information and Electrical Engineering, Shandong Jianzhu University Jinan 250101;
2. Research Institute of Wood Industry, CAF Beijing 100091
• Received:2018-04-02 Revised:2018-08-30 Online:2018-11-25 Published:2018-12-04

Abstract: [Objective] In order to satisfy the surface defect recognition quickly with high accuracy, a classification model based on random forest(RF) algorithm is proposed in this study, which can be used for identifying the big shavings, glue spots, debris, oil pollution quickly and accurately on the surface of fibreboard.[Method] Obtaining 100 surface defect images of 4 800 mm×2 400 mm fibreboard, using the Otsu algorithm to realize image segmentation, features including area(S), length(L), the length-width tatio(OR), compactness(J), rectanglarity(P), circularity(O), mean value(u), standard deviation(σD), smoothness(σP),skewness(σS), kurtosis(σK) and root mean square value(σR) of defect area were extracted, these were used as the experimental data. The experimental data were used for constructing RF classifier, choosing 2/3 data and eight features randomly by bootstrap method to construct k decision trees, the RF classifier were consist of these trees. The number of k is determined by calculating the outside data of the bag(OOB) error rate. 100 fibreboards with the wood shavings, glue spots, debris, and oil pollution were used to test by RF classifier.[Result] While k=600, the lowest average OOB error rate of the RF classifier is 0.004, the recognition accuracy of test is 99%, and the recognition time of one fibreboard is 525 ms. Therefore, the RF classifier is superior to NN and SVM on recognition time and accuracy.[Conclusion] The study proves the classifier based on RF algorithm is more feasibility and superiority on defect identification of fibreboard surface, it can achieve surface defect recognition quickly with high accuracy, satisfy the needs of on-line defect detection system of fibreboard.


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International Journal of Mathematics

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## MATH 4260 -数值分析代考| Numerical Analysis代写| 线性代数代写

##### 以下是MATH 4260 – Numerical Analysis: Linear and Nonlinear Problems的考试案例的答案解析

Let $A \in \mathbb{R}^{6 \times 6}$ be a matrix with characteristic polynomial
$$p_{A}(\lambda):=\operatorname{det}(A-\lambda I)=(\lambda+4)(\lambda-1)(\lambda-2)(\lambda-3)\left(\lambda^{2}+1\right)$$
(a) [2 Points] Determine all eigenvalues of $A$.

(b) [3 Points] We apply the power method to $A$. Which eigenvalue does the method converge to and what is the the ratio of the convergence rate?

(c) [4 Points] We apply inverse iteration with shift $s=\frac{7}{4}$ to $A$. Which eigenvalue does the method converge to and what is the ratio of the convergence rate?
Solution: The eigenvalues of the shifted and inverted matrix

Solution: The eigenvalues of $A$ are the zeros of $p_{A}(\lambda):$
$$-4,1,2,3, i,-i$$
where $i=\sqrt{-1}$

The dominant eigenvalue of $A$ is $-4,$ and the second largest in absolute value is $3 .$ Thus the power method converges to the eigenvalue -4 and
$$\text { ratio }=\frac{3}{4}$$

The eigenvalues of the shifted and inverted matrix
$$(A-s I)^{-1}=\left(A-\frac{7}{4} I\right)^{-1}$$
are
$$\frac{1}{-4-\frac{7}{4}}, \quad \frac{1}{1-\frac{7}{4}}, \quad \frac{1}{2-\frac{7}{4}}, \quad \frac{1}{3-\frac{7}{4}}, \quad \frac{1}{1-\frac{7}{4}}, \quad \frac{1}{-\mathrm{i}-\frac{7}{4}}$$
The dominant eigenvalue of $(A-s I)^{-1}$ is
$$\frac{1}{2-\frac{7}{4}}=4$$
and the second largest in absolute value is
$$\frac{1}{1-\frac{7}{4}}=-\frac{4}{3}$$
Thus inverse iteration converges to the eigenvalue 2 of $A$ and
ratio $=\frac{1}{3}$.

Let
$$A=\left[\begin{array}{rc} 1 & -1+\delta \\ -1 & 1 \end{array}\right]$$
where $\delta \in \mathbb{R}$ is a parameter.
(a) [2 Points] Show that $A$ is nonsingular if, and only if, $\delta \neq 0$. Determine $A^{-1}$ for $\delta \neq 0$.

(b) [4 Points] For $\delta \neq 0$, determine the condition number $\kappa_{1}(A)$ of $A$ with respect to the matrix norm $\|\cdot\|_{1}$

(c) [3 Points] Discuss the accuracy of the computed solution that you can expect when you solve linear systems $A x=b$ numerically.

Solution: Since
$$\operatorname{det} A=1 \cdot 1-(-1+\delta) \cdot(-1)=\delta$$
the matrix $A$ is nonsingular if, and only if, $\delta \neq 0$.
For $\delta \neq 0$
$$A^{-1}=\frac{1}{\operatorname{det} A} \operatorname{Adj}(A)=\frac{1}{\delta}\left[\begin{array}{cc} 1 & 1-\delta \\ 1 & 1 \end{array}\right]$$

Recall that $\|\cdot\|_{1}$ is the maximumabsolute column sum of the matrix. Thus, we have
$$\|A\|_{1}=\max \{2,|-1+\delta|+1\}=\left\{\begin{array}{ll} 2 & \text { if } 0<\delta \leq 2 \\ 1+|\delta-1| & \text { otherwise } \end{array}\right.$$
and
$$\left\|A^{-1}\right\|_{1}=\frac{1}{|\delta|} \max \{2,|1-\delta|+1\}=\left\{\begin{array}{ll} \frac{2}{|\delta|} & \text { if } 0<\delta \leq 2, \\ \frac{1+|\delta-1|}{|\delta|} & \text { otherwise. } \end{array}\right.$$
It follows that
$$\kappa_{1}(A)=\|A\|_{1} \cdot\left\|A^{-1}\right\|_{1}=\left\{\begin{array}{ll} \frac{4}{|\delta|} & \text { if } 0<\delta \leq 2, \\ \frac{(1+|\delta-1|)^{2}}{|\delta|} & \text { otherwise } \end{array}\right.$$

If $\kappa_{1}(A) \gg 1,$ the problem of solving linear systems $A x=b$ is ill-conditioned. In this case, we cannot expect to solve $A x=b$ numerically with any reasonable accuracy, no matter what algorithm we employ. The expression for $\kappa_{1}(A)$ obtained in (b) shows that $A x=b$ is ill-conditioned for $\delta \approx 0$ and for $|\delta| \gg 1$.

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##### MATH 4260 – Numerical Analysis: Linear and Nonlinear Problems代写线性代数代写

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## Generalized Inverses

Let $\mathbf{A}$ be an $m \times n$ matrix. A matrix $\mathbf{G}$ of order $n \times m$ is said to be a generalized inverse (or a g-inverse) of $\mathbf{A}$ if $\mathbf{A G A}=\mathbf{A}$.

If $\mathbf{A}$ is square and nonsingular, then $\mathbf{A}^{-\mathbf{1}}$ is the unique g-inverse of $\mathbf{A}$. Otherwise, A has infinitely many g-inverses, as we will see shortly.

1.1. Let $\mathbf{A}, \mathbf{G}$ be matrices of order $m \times n$ and $n \times m$ respectively. Then the following conditions are equivalent:
(i) $\mathbf{G}$ is a g-inverse of $\mathbf{A}$.
(ii) For any $\mathbf{y} \in \mathcal{C}(\mathbf{A}), \mathbf{x}=\mathbf{G} \mathbf{y}$ is a solution of $\mathbf{A} \mathbf{x}=\mathbf{y}$.

ProoF. (i) $\Rightarrow$ (ii). Any $\mathbf{y} \in \mathcal{C}(\mathbf{A})$ is of the form $\mathbf{y}=\mathbf{A z}$ for some $\mathbf{z}$. Then $\mathbf{A}(\mathbf{G y})=\mathbf{A G A z}=\mathbf{A z}=\mathbf{y}$
(ii) $\Rightarrow$ (i). Since $\mathbf{A G y}=\mathbf{y}$ for any $\mathbf{y} \in \mathcal{C}(\mathbf{A})$ we have $\mathbf{A G A z}=\mathbf{A z}$ for all $\mathbf{z}$. In particular, if we let $\mathbf{z}$ be the $i$ th column of the identity matrix, then we see that the $i$ th columns of AGA and $\mathbf{A}$ are identical. Therefore, $\mathbf{A G A}=\mathbf{A}$.

Let $\mathbf{A}=\mathbf{B C}$ be a rank factorization. We have seen that $\mathbf{B}$ admits a left inverse $\mathbf{B}{\ell}^{-}$, and $\mathbf{C}$ admits a right inverse $\mathbf{C}{\mathbf{r}}^{-} .$Then $\mathbf{G}=\mathbf{C}{\mathbf{r}}^{-} \mathbf{B}{\ell}^{-}$is a g-inverse of $\mathbf{A}$, since
$$\mathbf{A G A}=\mathbf{B C}\left(\mathbf{C}{\mathbf{r}}^{-} \mathbf{B}{\ell}^{-}\right) \mathbf{B C}=\mathbf{B C}=\mathbf{A}$$

Alternatively, if $\mathbf{A}$ has rank $r$, then by $7.3$ of Chapter 1 there exist nonsingular matrices $\mathbf{P}, \mathbf{Q}$ such that
$$\mathbf{A}=\mathbf{P}\left[\begin{array}{cc} \mathbf{I}{r} & \mathbf{0} \ \mathbf{0} & \mathbf{0} \end{array}\right] \mathbf{Q}$$ It can be verified that for $a n y \mathbf{U}, \mathbf{V}, \mathbf{W}$ of appropriate dimensions, $$\left[\begin{array}{cc} \mathbf{I}{r} & \mathbf{U} \ \mathbf{V} & \mathbf{W} \end{array}\right]$$
is a g-inverse of
$$\left[\begin{array}{ll} \mathbf{I}{r} & \mathbf{0} \ \mathbf{0} & \mathbf{0} \end{array}\right]$$ Then $$\mathbf{G}=\mathbf{Q}^{-1}\left[\begin{array}{cc} \mathbf{I}{r} & \mathbf{U} \ \mathbf{V} & \mathbf{W} \end{array}\right] \mathbf{P}^{-1}$$
is a g-inverse of $\mathbf{A}$. This also shows that any matrix that is not a square nonsingular matrix admits infinitely many g-inverses.

Another method that is particularly suitable for computing a g-inverse is as follows. Let $\mathbf{A}$ be of rank $r$. Choose any $r \times r$ nonsingular submatrix of $\mathbf{A}$. For convenience let us assume
$$\mathbf{A}=\left[\begin{array}{ll} \mathbf{A}{11} & \mathbf{A}{12} \ \mathbf{A}{21} & \mathbf{A}{22} \end{array}\right]$$

## Eigenvalues and the Spectral Theorem

Let $\mathbf{A}$ be an $n \times n$ matrix. The determinant $|\mathbf{A}-\lambda \mathbf{I}|$ is a polynomial in the (complex) variable $\lambda$ of degree $n$ and is called the characteristic polynomial of $\mathbf{A}$. The equation
$$|\mathbf{A}-\lambda \mathbf{I}|=0$$
is called the characteristic equation of $\mathbf{A}$. By the fundamental theorem of algebra, the equation has $n$ roots, and these roots are called the eigenvalues of $\mathbf{A}$.

The eigenvalues may not all be distinct. The number of times an eigenvalue occurs as a root of the characteristic equation is called the algebraic multiplicity of the eigenvalue.
We factor the characteristic polynomial as
$$|\mathbf{A}-\lambda \mathbf{I}|=\left(\lambda_{1}-\lambda\right) \cdots\left(\lambda_{n}-\lambda\right)$$
Setting $\lambda=0$ in (4) we see that $|\mathbf{A}|$ is just the product of the eigenvalues of $\mathbf{A}$. Similarly by equating the coefficient of $\lambda^{n-1}$ on either side of (4) we see that the trace of $\mathbf{A}$ equals the sum of the eigenvalues.
A principal submatrix of a square matrix is a submatrix formed by a set of rows and the corresponding set of columns. A principal minor of $\mathbf{A}$ is the determinant of a principal submatrix.
A square matrix $\mathbf{A}$ is called symmetric if $\mathbf{A}=\mathbf{A}^{\prime} .$ An $n \times n$ matrix $\mathbf{A}$ is said to be positive definite if it is symmetric and if for any nonzero vector $\mathbf{x}, \mathbf{x}^{\prime} \mathbf{A} \mathbf{x}>0$
The identity matrix is clearly positive definite and so is a diagonal matrix with only positive entries along the diagonal.

If $\mathbf{A}$ is positive definite, then it is nonsingular.

PROOF. If $\mathbf{A} \mathbf{x}=\mathbf{0}$, then $\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}=0$, and since $\mathbf{A}$ is positive definite, $\mathbf{x}=\mathbf{0}$ Therefore, A must be nonsingular.

The next result is obvious from the definition.

For $0 \leq \alpha \leq 1$, define

For $0 \leq \alpha \leq 1$, define
$$f(\alpha)=|\alpha \mathbf{A}+(1-\alpha) \mathbf{I}|$$

$\alpha \mathbf{A}+(1-\alpha) \mathbf{I}$ is positive definite, and therefore by 8.1, $f(\alpha) \neq 0,0 \leq$ $\alpha \leq 1$. Clearly, $f(0)=1$, and since $f$ is continuous, $f(1)=|\mathbf{A}|>0$

If $\mathbf{A}$ is positive definite, then any principal submatrix of $\mathbf{A}$ is positive definite.

ProoF. Since $\mathbf{A}$ is positive definite, $\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}>0$ for all $\mathbf{x} \neq \mathbf{0}$. Apply this condition to the set of vectors that have zeros in coordinates $j_{1}, \ldots, j_{s} .$ For such a vector $\mathbf{x}, \mathbf{x}^{\prime} \mathbf{A} \mathbf{x}$ reduces to an expression of the type $\mathbf{y}^{\prime} \mathbf{B y}$ where $\mathbf{B}$ is the principal submatrix of A formed by deleting rows and columns $j_{1}, \ldots, j_{s}$ from $\mathbf{A}$. It follows that $\mathbf{B}$, and similarly any principal submatrix of $\mathbf{A}$, is positive definite.

A symmetric $n \times n$ matrix $\mathbf{A}$ is said to be positive semidefinite if $\mathbf{x}^{\prime} \mathbf{A} \mathbf{x} \geq 0$ for all $\mathbf{x} \in R^{n}$

If $\mathbf{A}$ is a symmetric matrix, then the eigenvalues of $\mathbf{A}$ are all real.

PROOF. Suppose $\mu$ is an eigenvalue of $\mathbf{A}$ and let $\mu=\alpha+i \beta$, where $\alpha, \beta$ are real and $i=\sqrt{-1}$. Since $|\mathbf{A}-\mu \mathbf{I}|=0$, we have
$$|(\mathbf{A}-\alpha \mathbf{I})-i \beta \mathbf{I}|=0$$
Taking the complex conjugate of the above determinant and multiplying the two, we get
$$|(\mathbf{A}-\alpha \mathbf{I})-i \beta \mathbf{I} |(\mathbf{A}-\alpha \mathbf{I})+i \beta \mathbf{I}|=0$$
Thus
$$\left|(\mathbf{A}-\alpha \mathbf{I})^{2}+\beta^{2} \mathbf{I}\right|=0$$
Since $\mathbf{A}$ is symmetric, it is true that $\mathbf{A}^{2}$ is positive semidefinite (it follows from the definition that $\mathbf{B B}^{\prime}$ is positive semidefinite for any matrix $\left.\mathbf{B}\right)$. Thus if $\beta \neq 0$, then $\left|(\mathbf{A}-\alpha \mathbf{I})^{2}+\beta^{2} \mathbf{I}\right|$ is positive definite, and then by $\mathbf{8 . 1},(5)$ cannot hold. Thus $\beta=0$, and $\mu$ must be real.

## Frobenius Inequality

Let $\mathbf{B}$ be an $m \times r$ matrix of rank $r$. Then there exists a matrix $\mathbf{X}$ (called a left inverse of B), such that $\mathbf{X B}=\mathbf{I}$.

PROOF. If $m=r$, then $\mathbf{B}$ is nonsingular and admits an inverse. So suppose $r<m$. The columns of $\mathbf{B}$ are linearly independent. Thus we can find a set of $m-r$ columns that together with the columns of $\mathbf{B}$ form a basis for $R^{m}$. In other words, we can find a matrix $\mathbf{U}$ of order $m \times(m-r)$ such that $[\mathbf{B}, \mathbf{U}]$ is nonsingular. Let the inverse of $[\mathbf{B}, \mathbf{U}]$ be partitioned as $\left[\begin{array}{l}\mathbf{X} \ \mathbf{V}\end{array}\right]$, where $\mathbf{X}$ is $r \times m$. Since
$$\left[\begin{array}{l} \mathbf{X} \ \mathbf{V} \end{array}\right][\mathbf{B}, \mathbf{U}]=\mathbf{I}$$
we have $\mathbf{X B}=\mathbf{I}$
We can similarly show that an $r \times n$ matrix $\mathbf{C}$ of rank $r$ has a right inverse, i.e., a matrix $\mathbf{Y}$ such that $\mathbf{C Y}=\mathbf{I}$. Note that a left inverse or a right inverse is not unique, unless the matrix is square and nonsingular.

We can similarly show that an $r \times n$ matrix $\mathbf{C}$ of rank $r$ has a right inverse, i.e., a matrix $\mathbf{Y}$ such that $\mathbf{C Y}=\mathbf{I}$. Note that a left inverse or a rig

Let $\mathbf{B}$ be an $m \times r$ matrix of rank $r$. Then there exists a nonsingular matrix P such that
$$\mathbf{P B}=\left[\begin{array}{l} \mathbf{I} \ \mathbf{0} \end{array}\right]$$

ht inverse is not unique, unless the matrix is square and nonsingular.

Similarly, if $\mathbf{C}$ is $r \times n$ of rank $r$, then there exists a nonsingular matrix $\mathbf{Q}$ such that $\mathbf{C Q}=[\mathbf{I}, \mathbf{0}]$. These two results and the rank factorization (see $\mathbf{4 . 3})$ immediately lead to the following.

## Nonsingularity

Suppose we have $m$ linear equations in the $n$ unknowns $x_{1}, \ldots, x_{n}$. The equations can conveniently be expressed as a single matrix equation $\mathbf{A} \mathbf{x}=\mathbf{b}$, where $\mathbf{A}$ is the $m \times n$ matrix of coefficients. The equation $\mathbf{A} \mathbf{x}=\mathbf{b}$ is said to be consistent if it has at least one solution; otherwise, it is inconsistent. The equation is homogeneous if $\mathbf{b}=\mathbf{0}$. The set of solutions of the homogeneous equation $\mathbf{A x}=\mathbf{0}$ is clearly the null space of $\mathbf{A}$.
If the equation $\mathbf{A x}=\mathbf{b}$ is consistent, then we can write
$$\mathbf{b}=x_{1}^{0} \mathbf{a}{\mathbf{1}}+\cdots+x{n}^{0} \mathbf{a}{\mathbf{n}}$$ for some $x{1}^{0}, \ldots, x_{n}^{0}$, where $\mathbf{a}{1}, \ldots, \mathbf{a}{\mathbf{n}}$ are the columns of $\mathbf{A}$. Thus $\mathbf{b} \in \mathcal{C}(\mathbf{A})$ Conversely, if $\mathbf{b} \in \mathcal{C}(\mathbf{A})$, then $\mathbf{A x}=\mathbf{b}$ must be consistent. If the equation is consistent and if $\mathbf{x}^{0}$ is a solution of the equation, then the set of all solutions of the equation is given by
$$\left{\mathbf{x}^{\mathbf{0}}+\mathbf{x}: \mathbf{x} \in \mathcal{N}(\mathbf{A})\right}$$
Clearly, the equation $\mathbf{A x}=\mathbf{b}$ has either no solution, a unique solution, or infinitely many solutions.

A matrix $\mathbf{A}$ of order $n \times n$ is said to be nonsingular if $R(\mathbf{A})=n$; otherwise, the matrix is singular.

## Orthogonality

Let $S$ be a vector space. A function that assigns a real number $\langle\mathbf{x}, \mathbf{y}\rangle$ to every pair of vectors $\mathbf{x}, \mathbf{y}$ in $S$ is said to be an inner product if it satisfies the following conditions:

(i) $\langle\mathbf{x}, \mathbf{y}\rangle=\langle\mathbf{y}, \mathbf{x}\rangle$
(ii) $\langle\mathbf{x}, \mathbf{x}\rangle \geq 0$ and equality holds if and only if $\mathbf{x}=\mathbf{0}$.
(iii) $\langle c \mathbf{x}, \mathbf{y}\rangle=c\langle\mathbf{x}, \mathbf{y}\rangle .$
(iv) $\langle\mathbf{x}+\mathbf{y}, \mathbf{z}\rangle=\langle\mathbf{x}, \mathbf{z}\rangle+\langle\mathbf{y}, \mathbf{z}\rangle$
In $R^{n},\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\prime} \mathbf{y}=x_{1} y_{1}+\cdots+x_{n} y_{n}$ is easily seen to be an inner product. We will work with this inner product while dealing with $R^{n}$ and its subspaces, unless indicated otherwise.

For a vector $\mathbf{x}$, the positive square root of the inner product $\langle\mathbf{x}, \mathbf{x}\rangle$ is called the norm of $\mathbf{x}$, denoted by $|\mathbf{x}|$. Vectors $\mathbf{x}, \mathbf{y}$ are said to be orthogonal or perpendicular if $\langle\mathbf{x}, \mathbf{y}\rangle=0$, in which case we write $\mathbf{x} \perp \mathbf{y}$.

5.1. If $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{m}}$ are pairwise orthogonal nonzero vectors, then they are linearly independent.
PROOF. $\quad$ Suppose $c_{1} \mathbf{x}{\mathbf{1}}+\cdots+c{m} \mathbf{x}{\mathbf{m}}=\mathbf{0}$. Then $$\left\langle c{1} \mathbf{x}{\mathbf{1}}+\cdots+c{m} \mathbf{x}{\mathbf{m}}, \mathbf{x}{\mathbf{1}}\right\rangle=0$$
and hence,
$$\sum_{i=1}^{m} c_{i}\left\langle\mathbf{x}{\mathbf{i}}, \mathbf{x}{\mathbf{1}}\right\rangle=0$$
Since the vectors $\mathbf{x}{\mathbf{1}}, \ldots, \mathbf{x}{\mathbf{m}}$ are pairwise orthogonal, it follows that $c_{1}\left\langle\mathbf{x}{\mathbf{1}}, \mathbf{x}{\mathbf{1}}\right\rangle=0$ and since $\mathbf{x}{\mathbf{1}}$ is nonzero, $c{1}=0 .$ Similarly, we can show that each $c_{i}$ is zero. Therefore, the vectors are linearly independent.

A set of vectors $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{m}}$ is said to form an orthonormal basis for the vector space $S$ if the set is a basis for $S$ and furthermore, $\left\langle\mathbf{x}{\mathbf{i}}, \mathbf{x}{\mathbf{j}}\right\rangle$ is 0 if $i \neq j$ and 1 if $i=j$

We now describe the Gram-Schmidt procedure, which produces an orthonormal basis starting with a given basis $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{n}}$
Set $\mathbf{y}{\mathbf{1}}=\mathbf{x}{\mathbf{1}} .$ Having defined $\mathbf{y}{1}, \ldots, \mathbf{y}{\mathbf{i}-\mathbf{1}}$, we define
$$\mathbf{y}{\mathbf{i}}=\mathbf{x}{\mathbf{i}}-a_{i, i-1} \mathbf{y}{\mathbf{i}-\mathbf{1}}-\cdots-a{i 1} \mathbf{y}{\mathbf{1}}$$ where $a{i, i-1}, \ldots, a_{i 1}$ are chosen so that $\mathbf{y}{\mathbf{i}}$ is orthogonal to $\mathbf{y}{1}, \ldots, \mathbf{y}{\mathbf{i}-\mathbf{1}}$. Thus we must solve $\left\langle\mathbf{y}{\mathbf{i}}, \mathbf{y}{\mathbf{j}}\right\rangle=0, j=1, \ldots, i-1 .$ This leads to $$\left\langle\mathbf{x}{\mathbf{i}}-a_{i, i-1} \mathbf{y}{\mathbf{i}-\mathbf{1}}-\cdots-a{i 1} \mathbf{y}{\mathbf{1}}, \mathbf{y}{\mathbf{j}}\right\rangle=0, \quad j=1, \ldots, i-1$$

which gives
$$\left\langle\mathbf{x}{\mathbf{i}}, \mathbf{y}{\mathbf{j}}\right\rangle-\sum_{k=1}^{i-1} a_{i k}\left\langle\mathbf{y}{\mathbf{k}}, \mathbf{y}{\mathbf{j}}\right\rangle=0, \quad j=1, \ldots, i-1$$
Now, since $\mathbf{y}{1}, \ldots, \mathbf{y}{\mathbf{i}-\mathbf{1}}$ is an orthogonal set, we get
$$\left\langle\mathbf{x}{\mathbf{i}}, \mathbf{y}{\mathbf{j}}\right\rangle-a_{i j}\left\langle\mathbf{y}{\mathbf{j}}, \mathbf{y}{\mathbf{j}}\right\rangle=0$$

and hence,
$$a_{i j}=\frac{\left\langle\mathbf{x}{\mathbf{i}}, \mathbf{y}{\mathbf{j}}\right\rangle}{\left\langle\mathbf{y}{\mathbf{j}}, \mathbf{y}{\mathbf{j}}\right\rangle}, \quad j=1, \ldots, i-1$$
The process is continued to obtain the basis $\mathbf{y}{1}, \ldots, \mathbf{y}{\mathbf{n}}$ of pairwise orthogonal vectors. Since $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{n}}$ are linearly independent, each $\mathbf{y}{\mathbf{i}}$ is nonzero. Now if we set $\mathbf{z}{\mathbf{i}}=\frac{\mathbf{y}{\mathrm{i}}}{\left|\mathbf{y}{\mathrm{i}}\right|}$, then $\mathbf{z}{1}, \ldots, \mathbf{z}{\mathbf{n}}$ is an orthonormal basis. Note that the linear span of $\mathbf{z}{1}, \ldots, \mathbf{z}{\mathbf{i}}$ equals the linear span of $\mathbf{x}{\mathbf{1}}, \ldots, \mathbf{x}{\mathbf{i}}$ for each $i$

We remark that given a set of linearly independent vectors $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{m}}$, the GramSchmidt procedure described above can be used to produce a pairwise orthogonal set $\mathbf{y}{1}, \ldots, \mathbf{y}{\mathbf{m}}$, such that $\mathbf{y}{\mathbf{i}}$ is a linear combination of $\mathbf{x}{\mathbf{1}}, \ldots, \mathbf{x}_{\mathbf{i}-\mathbf{1}}, i=1, \ldots, m$ This fact is used in the proof of the next result.

Let $W$ be a set (not necessarily a subspace) of vectors in a vector space $S$. We define
$$W^{\perp}={\mathbf{x}: \mathbf{x} \in S,\langle\mathbf{x}, \mathbf{y}\rangle=0 \text { for all } \mathbf{y} \in W}$$
It follows from the definitions that $W^{\perp}$ is a subspace of $S$.

## Rank

Let $\mathbf{A}$ be an $m \times n$ matrix. The subspace of $R^{m}$ spanned by the column vectors of $\mathbf{A}$ is called the column space or the column span of $\mathbf{A}$ and is denoted by $\mathcal{C}(\mathbf{A})$. Similarly, the subspace of $R^{n}$ spanned by the row vectors of $\mathbf{A}$ is called the row space of $\mathbf{A}$, denoted by $\mathcal{R}(\mathbf{A})$. Clearly, $\mathcal{R}(\mathbf{A})$ is isomorphic to $\mathcal{C}\left(\mathbf{A}^{\prime}\right)$. The dimension

of the column space is called the column rank, whereas the dimension of the row space is called the row rank of the matrix. These two definitions turn out be very short-lived in any linear algebra book, since the two ranks are always equal, as we show in the next result.

The following operations performed on a matrix $\mathbf{A}$ are called elementary column operations:
(i) Interchange two columns of $\mathbf{A}$.
(ii) Multiply a column of $\mathbf{A}$ by a nonzero scalar.
(iii) Add a scalar multiple of one column to another column.
These operations clearly leave $\mathcal{C}(\mathbf{A})$ unaffected, and therefore they do not change the rank of the matrix. We can define elementary row operations similarly. The elementary row and column operations are particularly useful in computations. Thus to find the rank of a matrix we first reduce it to a matrix with several zeros by these operations and then compute the rank of the resulting matrix.

## Basis and Dimension

The linear span of (or the space spanned by) the vectors $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{m}}$ is defined to be the set of all linear combinations $c_{1} \mathbf{x}{\mathbf{1}}+\cdots+c{m} \mathbf{x}{\mathbf{m}}$ where $c{1}, \ldots, c_{m}$ are real numbers. The linear span is a subspace; this follows from the definition.

A set of vectors $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{m}}$ is said to be linearly dependent if there exist real numbers $c_{1}, \ldots, c_{m}$ such that at least one $c_{i}$ is nonzero and $c_{1} \mathbf{x}{\mathbf{1}}+\cdots+c{m} \mathbf{x}{\mathbf{m}}=\mathbf{0}$ A set is linearly independent if it is not linearly dependent. Strictly speaking, we should refer to a collection (or a multiset) of vectors rather than a set of vectors in the two preceding definitions. Thus when we talk of vectors $\mathbf{x}{1}, \ldots, \mathbf{x}_{\mathbf{m}}$ being linearly dependent or independent, we allow for the possibility of the vectors not necessarily being distinct.
The following statements are easily proved:
(i) The set consisting of the zero vector alone is linearly dependent.
(ii) If $X \subset Y$ and if $X$ is linearly dependent, then so is $Y$.
(iii) If $X \subset Y$ and if $Y$ is linearly independent, then so is $X$.
A set of vectors is said to form a basis for the vector space $S$ if it is linearly independent and its linear span equals $S$.

Let $\mathbf{e}{\mathbf{i}}$ be the $i$ th column of the $n \times n$ identity matrix. The set $\mathbf{e}{1}, \ldots, \mathbf{e}_{\mathbf{n}}$ forms a basis for $R^{n}$, called the standard basis.

If $\mathbf{x}{1}, \ldots, \mathbf{x}{\mathbf{m}}$ is a basis for $S$, then any vector $\mathbf{x}$ in $S$ admits a unique representation as a linear combination $c_{1} \mathbf{x}{\mathbf{1}}+\cdots+c{m} \mathbf{x}{\mathbf{m}}$. For if $$\mathbf{x}=c{1} \mathbf{x}{\mathbf{1}}+\cdots+c{m} \mathbf{x}{\mathbf{m}}=d{1} \mathbf{x}{\mathbf{1}}+\cdots+d{m} \mathbf{x}{\mathbf{m}}$$ then $$\left(c{1}-d_{1}\right) \mathbf{x}{\mathbf{1}}+\cdots+\left(c{m}-d_{m}\right) \mathbf{x}{\mathbf{m}}=\mathbf{0}$$ and since $\mathbf{x}{\mathbf{1}}, \ldots, \mathbf{x}{\mathbf{m}}$ are linearly independent, $c{i}=d_{i}$ for each $i$.
A vector space is said to be finite-dimensional if it has a basis consisting of finitely many vectors. The vector space containing only the zero vector is also finite-dimensional. We will consider only finite-dimensional vector spaces. Very often it will be implicitly assumed that the vector spaces under consideration are nontrivial, i.e., contain vectors other than the zero vector.

## Vector Spaces and Subspaces

A nonempty set $S$ is called a vector space if it satisfies the following conditions:
(i) For any $\mathbf{x}, \mathbf{y}$ in $S, \mathbf{x}+\mathbf{y}$ is defined and is in $S$. Furthermore,
\begin{aligned} \mathbf{x}+\mathbf{y} &=\mathbf{y}+\mathbf{x}, \quad \text { (commutativity) } \ \mathbf{x}+(\mathbf{y}+\mathbf{z}) &=(\mathbf{x}+\mathbf{y})+\mathbf{z} . \quad \text { (associativity) } \end{aligned}
(ii) There exists an element in $S$, denoted by $\mathbf{0}$, such that $\mathbf{x}+\mathbf{0}=\mathbf{x}$ for all $\mathbf{x}$.
(iii) For any $\mathbf{x}$ in $S$, there exists an element $\mathbf{y}$ in $S$ such that $\mathbf{x}+\mathbf{y}=\mathbf{0}$.
(iv) For any $\mathbf{x}$ in $S$ and any real number $c, c \mathbf{x}$ is defined and is in $S ;$ moreover, $1 \mathbf{x}=\mathbf{x}$ for any $\mathbf{x}$
(v) For any $\mathbf{x}{\mathbf{1}}, \mathbf{x}{\mathbf{2}}$ in $S$ and real numbers $c_{1}, c_{2}, c_{1}\left(\mathbf{x}{1}+\mathbf{x}{\mathbf{2}}\right)=c_{1} \mathbf{x}{\mathbf{1}}+c{1} \mathbf{x}{\mathbf{2}},\left(c{1}+\right.$ $\left.c_{2}\right) \mathbf{x}{\mathbf{1}}=c{1} \mathbf{x}{\mathbf{1}}+c{2} \mathbf{x}{\mathbf{1}}$ and $c{1}\left(c_{2} \mathbf{x}{\mathbf{1}}\right)=\left(c{1} c_{2}\right) \mathbf{x}_{\mathbf{1}}$

Elements in $S$ are called vectors. If $\mathbf{x}, \mathbf{y}$ are vectors, then the operation of taking their sum $\mathbf{x}+\mathbf{y}$ is referred to as vector addition. The vector in (ii) is called the $z$ ero vector. The operation in (iv) is called scalar multiplication. A vector space may be defined with reference to any field. We have taken the field to be the field of real numbers as this will be sufficient for our purpose.

The set of column vectors of order $n$ (or $n \times 1$ matrices) is a vector space. So is the set of row vectors of order $n$. These two vector spaces are the ones we consider most of the time.
Let $R^{n}$ denote the set $R \times R \times \cdots \times R$, taken $n$ times, where $R$ is the set of real numbers. We will write elements of $R^{n}$ either as column vectors or as row vectors depending upon whichever is convenient in a given situation.
If $S, T$ are vector spaces and $S \subset T$, then $S$ is called a subspace of $T$. Let us describe all possible subspaces of $R^{3}$. Clearly, $R^{3}$ is a vector space, and so is the space consisting of only the zero vector, i.e., the vector of all zeros. Let $c_{1}, c_{2}, c_{3}$ be real numbers. The set of all vectors $\mathbf{x} \in R^{3}$ that satisfy
$$c_{1} x_{1}+c_{2} x_{2}+c_{3} x_{3}=0$$
is a subspace of $R^{3}$ (Here $x_{1}, x_{2}, x_{3}$ are the coordinates of $\left.\mathbf{x}\right)$. Geometrically, this set represents a plane passing through the origin. Intersection of two distinct planes through the origin is a straight line through the origin and is also a subspace. These are the only possible subspaces of $R^{3}$

## Vector Spaces and Matrices

We consider only real matrices. Although our treatment is self-contained, the reader is assumed to be familiar with basic operations on matrices. We also assume knowledge of elementary properties of the determinant.

An $m \times n$ matrix consists of $m n$ real numbers arranged in $m$ rows and $n$ columns. We denote matrices by bold letters. The entry in row $i$ and column $j$ of the matrix A is denoted by $a_{i j}$. An $m \times 1$ matrix is called a column vector of order $m$; similarly, a $1 \times n$ matrix is a row vector of order $n$. An $m \times n$ matrix is called a square matrix if $m=n$

If $\mathbf{A}, \mathbf{B}$ are $m \times n$ matrices, then $\mathbf{A}+\mathbf{B}$ is defined as the $m \times n$ matrix with $(i, j)$-entry $a_{i j}+b_{i j}$. If $\mathbf{A}$ is a matrix and $c$ is a real number, then $c \mathbf{A}$ is obtained by multiplying each element of $\mathbf{A}$ by $c$.

If $\mathbf{A}$ is $m \times p$ and $\mathbf{B}$ is $p \times n$, then their product $\mathbf{C}=\mathbf{A} \mathbf{B}$ is an $m \times n$ matrix with $(i, j)$-entry given by
$$c_{i j}=\sum_{k=1}^{p} a_{i k} b_{k j}$$
The following properties hold:
\begin{aligned} (\mathbf{A} \mathbf{B}) \mathbf{C} &=\mathbf{A}(\mathbf{B C}) \ \mathbf{A}(\mathbf{B}+\mathbf{C}) &=\mathbf{A} \mathbf{B}+\mathbf{A C} \ (\mathbf{A}+\mathbf{B}) \mathbf{C} &=\mathbf{A} \mathbf{C}+\mathbf{B C} \end{aligned}

The transpose of the $m \times n$ matrix $\mathbf{A}$, denoted by $\mathbf{A}^{\prime}$, is the $n \times m$ matrix whose $(i, j)$-entry is $a_{j i}$. It can be verified that $\left(\mathbf{A}^{\prime}\right)^{\prime}=\mathbf{A},(\mathbf{A}+\mathbf{B})^{\prime}=\mathbf{A}^{\prime}+\mathbf{B}^{\prime},(\mathbf{A} \mathbf{B})^{\prime}=$ $\mathbf{B}^{\prime} \mathbf{A}^{\prime}$

A good understanding of the definition of matrix multiplication is quite useful. We note some simple facts that are often required. We assume that all products occurring here are defined in the sense that the orders of the matrices make them compatible for multiplication.
(i) The $j$ th column of $\mathbf{A B}$ is the same as $\mathbf{A}$ multiplied by the $j$ th column of $\mathbf{B}$.
(ii) The $i$ th row of $\mathbf{A B}$ is the same as the $i$ th row of $\mathbf{A}$ multiplied by $\mathbf{B}$.
(iii) The $(i, j)$-entry of $\mathbf{A B C}$ is obtained as
$$\left(x_{1}, \ldots, x_{p}\right) \mathbf{B}\left[\begin{array}{c} y_{1} \ \vdots \ y_{q} \end{array}\right]$$
where $\left(x_{1}, \ldots, x_{p}\right)$ is the $i$ th row of $\mathbf{A}$ and $\left(y_{1}, \ldots, y_{q}\right)^{\prime}$ is the $j$ th column of C.
(iv) If $\mathbf{A}=\left[\mathbf{a}{1}, \ldots, \mathbf{a}{\mathbf{n}}\right]$ and
$$\mathbf{B}=\left[\begin{array}{c} \mathbf{b}{\mathbf{1}}^{\prime} \ \vdots \ \mathbf{b}{\mathbf{n}}^{\prime} \end{array}\right]$$
where $\mathbf{a}{\mathbf{i}}$ denote columns of $\mathbf{A}$ and $\mathbf{b}{\mathbf{j}}^{\prime}$ denote rows of $\mathbf{B}$, then
$$\mathbf{A} \mathbf{B}=\mathbf{a}{1} \mathbf{b}{\mathbf{1}}^{\prime}+\cdots+\mathbf{a}{\mathbf{n}} \mathbf{b}{\mathbf{n}}^{\prime}$$

A diagonal matrix is a square matrix $\mathbf{A}$ such that $a_{i j}=0, i \neq j$. We denote the diagonal matrix
$$\left[\begin{array}{cccc} \lambda_{1} & 0 & \cdots & 0 \ 0 & \lambda_{2} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & \lambda_{n} \end{array}\right]$$
by $\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right)$. When $\lambda_{i}=1$ for all $i$, this matrix reduces to the identity matrix of order $n$, which we denote by $\mathbf{I}{\mathbf{n}}$, or often simply by $\mathbf{I}$ if the order is clear from the context. Observe that for any square matrix $\mathbf{A}$, we have $\mathbf{A I}=\mathbf{I A}=\mathbf{A}$ The entries $a{11}, \ldots, a_{n n}$ are said to constitute the (main) diagonal entries of $\mathbf{A}$. The trace of $\mathbf{A}$ is defined as
$$\operatorname{trace} \mathbf{A}=a_{11}+\cdots+a_{n n}$$
It follows from this definition that if $\mathbf{A}, \mathbf{B}$ are matrices such that both $\mathbf{A B}$ and $\mathbf{B A}$ are defined, then
$$\operatorname{trace} \mathbf{A} \mathbf{B}=\operatorname{trace} \mathbf{B} \mathbf{A}$$

The determinant of an $n \times n$ matrix $\mathbf{A}$, denoted by $|\mathbf{A}|$, is defined as
$$|\mathbf{A}|=\sum_{\sigma} \epsilon(\sigma) a_{1 \sigma(1)} \cdots a_{n \sigma(n)}$$
where the summation is over all permutations ${\sigma(1), \ldots, \sigma(n)}$ of ${1, \ldots, n}$ and $\epsilon(\sigma)$ is 1 or $-1$ according as $\sigma$ is even or odd.
We state some basic properties of the determinant without proof:
(i) The determinant can be evaluated by expansion along a row or a column. Thus, expanding along the first row,
$$|\mathbf{A}|=\sum_{j=1}^{n}(-1)^{1+j} a_{1 j}\left|\mathbf{A}{\mathbf{1 j}}\right|$$ where $\mathbf{A}{1 j}$ is the submatrix obtained by deleting the first row and the $j$ th column of $\mathbf{A}$. We also note that
$$\sum_{j=1}^{n}(-1)^{1+j} a_{i j}\left|\mathbf{A}_{\mathbf{1 j}}\right|=0, \quad i=2, \ldots, n$$

(ii) The determinant changes sign if two rows (or columns) are interchanged.
(iii) The determinant is unchanged if a constant multiple of one row is added to another row. A similar property is true for columns.
(iv) The determinant is a linear function of any column (row) when all the other columns (rows) are held fixed.
(v) $|\mathbf{A} \mathbf{B}|=|\mathbf{A}||\mathbf{B}|$
The matrix $\mathbf{A}$ is upper triangular if $a_{i j}=0, i>j$. The transpose of an upper triangular matrix is lower triangular.

It will often be necessary to work with matrices in partitioned form. For example, let
$$\mathbf{A}=\left[\begin{array}{ll} \mathbf{A}{11} & \mathbf{A}{12} \ \mathbf{A}{21} & \mathbf{A}{22} \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{ll} \mathbf{B}{11} & \mathbf{B}{12} \ \mathbf{B}{21} & \mathbf{B}{22} \end{array}\right]$$
be two matrices where each $\mathbf{A}{\mathbf{i j}}, \mathbf{B}{\mathbf{i j}}$ is itself a matrix. If compatibility for matrix multiplication is assumed throughout (in which case, we say that the matrices are partitioned conformally), then we can write
$$\mathbf{A} \mathbf{B}=\left[\begin{array}{ll} \mathbf{A}{11} \mathbf{B}{11}+\mathbf{A}{12} \mathbf{B}{21} & \mathbf{A}{11} \mathbf{B}{12}+\mathbf{A}{12} \mathbf{B}{22} \ \mathbf{A}{21} \mathbf{B}{11}+\mathbf{A}{22} \mathbf{B}{21} & \mathbf{A}{21} \mathbf{B}{12}+\mathbf{A}{22} \mathbf{B}{22} \end{array}\right]$$

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## HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

Moderator: SecondMan

Ayush Bakshi
Posts: 15
Joined: Wed Jan 23, 2019 8:59 am
Been thanked: 2 times

### HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

It's draining my energy (and possibly sanity) but I can't figure out: (Images attached down there ↓)

1. How to use garbage matte with Ultra Keyer?
I attached B-spline's node to Ultra Keyer's 'Garbage Matte'. It worked half expectedly (literally). It didn't performed keying operation in the garbage mask area (as expected) but it didn't masked the entire thing... I just wanted that part masked out
It did worked as expected when I check 'Post multiply image' but it gave black edges.
Also, It (spline node connected to 'Garbage Matte') behaved normally with Delta keyer.

2. How to connect splines' position to tracker without moving the spline?
I set spline → Happy with the shape and position → Tracked something using 'Tracker' node → Connected spline's 'Center' to tracker's 'Offset position' (right decision?). The problem is that this operation moved the spline which I don't want. I can reposition it by selecting all the anchor points but it's not easy to redo when the shape is complex and you already spent time in positioning every single point.
Also what's the difference between all those 'connect to' options (refer to attached images).

What am I doing wrong? Just want to keep my life simple but Fusion is trickier than I anticipated for :'(
There are so few tutorials for BMD Fusion (comparatively speaking). In future I'll be using Nuke because BMD fusion isn't used much in Industries (in my country). But I've to use it because it's in my course [crying with tears emoji]

Images↓

It also crashes randomly in an instant but that's the problem for another thread...

Movalex
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### Re: HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

1. Uncheck Subtract background when premultiplying Ultrakeyer or you'll have double premultiplication. Also try Deltakeyer, I like it better.
2. Tracker Plus from Reactor has Set/Change reference time setting. Probably it could help.

UPD:
a) install Eyeon Legacy scripts from Reactor.
b) select tracker on the frame you need
c) add spline on the same frame and right click - Script- Eyeon Legacy -- Attach Mask... and choose current Tracker Path
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Ayush Bakshi
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### Re: HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

Movalex wrote:
Wed Feb 13, 2019 2:54 am
a) install Eyeon Legacy scripts from Reactor.
b) select tracker on the frame you need
c) add spline on the same frame and right click - Script- Eyeon Legacy -- Attach Mask... and choose current Tracker Path
Thanks! Ultra Keyer Problem Solved.

I did what you said and applied that script to Transform node. It worked. 7 Splines are connected in chain and on the end is Transform node. Splines are moving with it (as expected) BUT.. When I connected 'Transform node OUT' to Delta Keyer's Solid matte... It works only for first frame.. After that splines doesn't affect Solid matte as if they aren't there. If I don't introduce Transform node then things work fine.

Image ↓

Movalex
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### Re: HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

Did you try Flatten Transform? Also check if it is a memory issue (purge cache)

Ayush Bakshi
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### Re: HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

Movalex wrote:
Wed Feb 13, 2019 6:21 am
Did you try Flatten Transform? Also check if it is a memory issue (purge cache)
I tried both. Nothing changed.
If someone want to achieve this then is it the right approach? Logically speaking... I can't understand why it's not working.

SirEdric
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### Re: HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

So you're piping the RGB out of the Tracker into the Garbagemask input?
While in theory this should work, what happens if you add a BitmapMask to the GarbageInout first and pipe the XF into BitmapMask?

Added in 3 minutes 5 seconds:
And/or (just to be ultra-safe to not mix up RGB/Mask pipes) something like this:

Code: Select all

{
Tools = ordered() {
DrawMode = "InsertAndModify",
DrawMode2 = "InsertAndModify",
Inputs = {
MaskWidth = Input { Value = 1920, },
MaskHeight = Input { Value = 1080, },
PixelAspect = Input { Value = { 1, 1 }, },
ClippingMode = Input { Value = FuID { "None" }, },
Polyline = Input {
SourceOp = "Polygon1Polyline",
Source = "Value",
},
Polyline2 = Input {
Value = Polyline {
},
Disabled = true,
},
},
ViewInfo = OperatorInfo { Pos = { 55, -16.5 } },
},
Polygon1Polyline = BezierSpline {
SplineColor = { Red = 173, Green = 255, Blue = 47 },
NameSet = true,
KeyFrames = {
[0] = { 0, Flags = { Linear = true, LockedY = true }, Value = Polyline {
Closed = true,
Points = {
{ X = -0.373333334922791, Y = -0.109467454254627, LX = -0.0175039738414455, LY = 0.0854483930129737, RX = 0.0166563707029685, RY = -0.081310685384314 },
{ X = -0.270000010728836, Y = -0.292899399995804, LX = -0.0400000107288361, LY = 0.0295858071047879, RX = 0.0613124512231215, RY = -0.0453494467115552 },
{ X = -0.0799999982118607, Y = -0.186390534043312, LX = 0.00666666845480602, LY = -0.0769230784220103, RX = -0.0108282208060373, RY = 0.124940978223266 },
{ X = -0.16333332657814, Y = 0.162721887230873, LX = 0.0433333400885264, LY = -0.0295858050768192, RX = -0.054983657387011, RY = 0.0375400503755176 },
{ X = -0.340000013113022, Y = 0.156804733286948, LX = 0.0471298661672112, LY = 0.0627085244656244, RX = -0.0409847655616014, RY = -0.0545321763660259 }
}
} }
}
},
DrawMode = "InsertAndModify",
DrawMode2 = "InsertAndModify",
Inputs = {
MaskWidth = Input { Value = 1920, },
MaskHeight = Input { Value = 1080, },
PixelAspect = Input { Value = { 1, 1 }, },
ClippingMode = Input { Value = FuID { "None" }, },
Polyline = Input {
SourceOp = "Polygon2Polyline",
Source = "Value",
},
Polyline2 = Input {
Value = Polyline {
},
Disabled = true,
},
SourceOp = "Polygon1",
}
},
ViewInfo = OperatorInfo { Pos = { 55, 16.5 } },
},
Polygon2Polyline = BezierSpline {
SplineColor = { Red = 96, Green = 255, Blue = 47 },
NameSet = true,
KeyFrames = {
[0] = { 0, Flags = { Linear = true, LockedY = true }, Value = Polyline {
Closed = true,
Points = {
{ Linear = true, X = 0.226666674017906, Y = 0.304733723402023, LX = 0.0566666672627131, LY = -0.0177514751752218, RX = -0.0122222246726354, RY = -0.092702168155704 },
{ Linear = true, X = 0.189999997615814, Y = 0.0266272183507681, LX = 0.0122222246726354, LY = 0.092702168155704, RX = 0.0433333341280619, RY = -0.0828402364916762 },
{ Linear = true, X = 0.319999992847443, Y = -0.221893489360809, LX = -0.0433333341280619, LY = 0.0828402364916762, RX = 0.0255555579397413, RY = 0.0650887568086327 },
{ Linear = true, X = 0.396666675806046, Y = -0.0266272183507681, LX = -0.0255555579397413, LY = -0.0650887568086327, RX = 0, RY = 0.0927021694305321 },
{ Linear = true, X = 0.396666675806046, Y = 0.251479297876358, LX = -0, LY = -0.0927021694305321, RX = -0.0566666672627131, RY = 0.0177514751752218 }
}
} }
}
},
Background1 = Background {
Inputs = {
Width = Input { Value = 1920, },
Height = Input { Value = 1080, },
["Gamut.SLogVersion"] = Input { Value = FuID { "SLog2" }, },
TopLeftRed = Input { Value = 1, },
TopLeftGreen = Input { Value = 1, },
TopLeftBlue = Input { Value = 1, },
SourceOp = "Polygon3",
}
},
ViewInfo = OperatorInfo { Pos = { 220, 49.5 } },
},
DrawMode = "InsertAndModify",
DrawMode2 = "InsertAndModify",
Inputs = {
MaskWidth = Input { Value = 1920, },
MaskHeight = Input { Value = 1080, },
PixelAspect = Input { Value = { 1, 1 }, },
ClippingMode = Input { Value = FuID { "None" }, },
Polyline = Input {
SourceOp = "Polygon3Polyline",
Source = "Value",
},
Polyline2 = Input {
Value = Polyline {
},
Disabled = true,
},
SourceOp = "Polygon2",
}
},
ViewInfo = OperatorInfo { Pos = { 55, 49.5 } },
},
Polygon3Polyline = BezierSpline {
SplineColor = { Red = 96, Green = 255, Blue = 47 },
NameSet = true,
KeyFrames = {
[0] = { 0, Flags = { Linear = true, LockedY = true }, Value = Polyline {
Closed = true,
Points = {
{ Linear = true, X = 0.0333333350718021, Y = 0.204142019152641, LX = 0.0444444455206394, LY = -0.14595661063989, RX = -0.00888888946837848, RY = -0.106508878179086 },
{ Linear = true, X = 0.00666666682809591, Y = -0.115384615957737, LX = 0.00888888946837848, LY = 0.106508878179086, RX = 0.0333333332795236, RY = -0.092702169434206 },
{ Linear = true, X = 0.106666669249535, Y = -0.393491119146347, LX = -0.0333333332795236, LY = 0.092702169434206, RX = 0.0222222213612662, RY = 0.0236686373485851 },
{ Linear = true, X = 0.173333331942558, Y = -0.322485208511353, LX = -0.0222222213612662, LY = -0.0236686373485851, RX = -0.00222222175863055, RY = 0.0295857992868217 },
{ Linear = true, X = 0.16666667163372, Y = -0.233727812767029, LX = 0.00222222175863055, LY = -0.0295857992868217, RX = -0.0444444455206394, RY = 0.14595661063989 }
}
} }
}
},
Transform1 = Transform {
Inputs = {
Input = Input {
SourceOp = "Background1",
Source = "Output",
},
},
ViewInfo = OperatorInfo { Pos = { 385, 49.5 } },
},
Inputs = {
MaskWidth = Input { Value = 1920, },
MaskHeight = Input { Value = 1080, },
PixelAspect = Input { Value = { 1, 1 }, },
ClippingMode = Input { Value = FuID { "None" }, },
Image = Input {
SourceOp = "Transform1",
Source = "Output",
},
},
ViewInfo = OperatorInfo { Pos = { 550, 49.5 } },
},
UltraKeyer1 = UltraKeyer {
CtrlWZoom = false,
Inputs = {
BackgroundRed = Input { Value = 0.5, },
BackgroundGreen = Input { Value = 0.5, },
BackgroundBlue = Input { Value = 0.5, },
["Garbage.Matte"] = Input {
SourceOp = "Bitmap1",
},
PostMultiplyImage = Input { Value = 0, },
},
ViewInfo = OperatorInfo { Pos = { 715, 49.5 } },
}
}
}

intelligent machine
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### Re: HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

To add my hunch: if you are referring to the solid matte and not garbage matte input, make sure that "Alpha" is turned ON on the blending tab of the transform tool:

Ayush Bakshi
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### Re: HELP!! Tracking, Connecting to Tracker And Garbage matte on Ultra Keyer

I did the obvious workaround and applied tracker data to each of the 7 splines using the script suggested by Movalex that keeps the anchor point's position. But... it's still a workaround. I didn't tried SirEdric's and intelligent_machine's methods yet.. (since the job is done I ain't feeling motivated ).
Thanks! Everyone. A lot is left on the table I must finish as soon as I could for the final assignment that is coming soon (adding one's 3D animated model in real footage)


text:
2022
Ref: SCART-2022-0104

Seismic detection of a deep mantle discontinuity within Mars by InSight

Huang, Quancheng ; Schmerr, Nicholas C. ; King, Scott D. ; Kim, Doyeon ; Rivoldini, Attilio ; Plesa, Ana-Catalina ; Samuel, Henri ; Maguire, Ross R. ; Karakostas, Foivos ; Lekić, Vedran ; Charalambous, Constantinos ; Collinet, Max ; Myhill, Robert ; Antonangeli, Daniele ; Drilleau, Mélanie ; Bystricky, Misha ; Bollinger, Caroline ; Michaut, Chloé ; Gudkova, Tamara ; Irving, Jessica C. E. ; Horleston, Anna ; Fernando, Benjamin ; Leng, Kuangdai ; Nissen-Meyer, Tarje ; Bejina, Frederic

published in Proceedings of the National Academy of Sciences, 119 issue 42 (2022)

Abstract: Constraining the thermal and compositional state of the mantle is crucial for deciphering the formation and evolution of Mars. Mineral physics predicts that Mars? deep mantle is demarcated by a seismic discontinuity arising from the pressure-induced phase transformation of the mineral olivine to its higher-pressure polymorphs, making the depth of this boundary sensitive to both mantle temperature and composition. Here, we report on the seismic detection of a midmantle discontinuity using the data collected by NASA?s InSight Mission to Mars that matches the expected depth and sharpness of the postolivine transition. In five teleseismic events, we observed triplicated P and S waves and constrained the depth of this discontinuity to be 1,006 $\pm$40 km by modeling the triplicated waveforms. From this depth range, we infer a mantle potential temperature of 1,605 $\pm$100 K, a result consistent with a crust that is 10 to 15 times more enriched in heat-producing elements than the underlying mantle. Our waveform fits to the data indicate a broad gradient across the boundary, implying that the Martian mantle is more enriched in iron compared to Earth. Through modeling of thermochemical evolution of Mars, we observe that only two out of the five proposed composition models are compatible with the observed boundary depth. Our geodynamic simulations suggest that the Martian mantle was relatively cold 4.5 Gyr ago (1,720 to 1,860 K) and are consistent with a present-day surface heat flow of 21 to 24 mW/m2.

DOI: 10.1073/pnas.2204474119
Funding: 3PRPDPLANINT/3PRPDPLANINT/3PRPDPLANINT

The record appears in these collections:
Royal Observatory of Belgium > Reference Systems & Planetology
Science Articles > Peer Reviewed Articles