thank you for the kind introduction and
inviting me for this talk and good
evening to all the participants and all
the august audience over here
digitally definitely yes as writing
mentioned earlier in the introduction as
well so today I'm going to talk about on
complex eigenvalues especially and I
have given the title of the talk as
complex is easy again complex
eigenvalues and they're intuitive
meaning with their applications in
science so I'll start this whole
discussion to make it relatively more
interactive although I'm not available
here physically because of the physical
boundaries definitely we all are aware
of but I will take this opportunity to
make it as much as interactive as
possible
and definitely look forward for your
responses on my LinkedIn and definitely
to the the organizers but to begin with
I'll start with a small anecdote why I
chose this topic definitely I can values
in journal Applied Mathematics and
Mathematical modeling with the
theme of this particular I would say
digital conference or confluence is a
mathematical, Multi-disciplinary application in
design so during my PhD days I mean it
was a question which somebody one of my
friends asked me that do you know what
is the physical significance or the
geometrical meaning of eigenvalue and in
particular complex eigenvalues and
we used to deal with eigenvalues a lot
but to actually get into the intricacy
to make it like offhand
you know solution and to tell somebody
that what is eigenvalue and geometrical
interpretation of that especially in the
complex domain is what actually always
excited me there on so that's why I
chose this topic with no further due let
me get into that as I promised to make
it more interactive so I'll share a few
questions the answers of that will be
there during the presentation
partly and I look forward for the
answers so that to make it more
interactive earlier many of you or maybe
all of you it's like we like to who
would like to interact with me
later on pass on these the answers and
their understanding through the LinkedIn
page of mine so the first question is
can you tell the famous algorithm that
utilizes eigenvectors and has
revolutionized the world wide web yeah
second is can you guess the worth yes I
underline this the worth of
eigenvectors in the year 2004 for google
yeah so please stay tuned and be
inquisitive about the answers hidden in
the presentation
let me start formally the presentation
with an agenda so to begin with I'll
talk about the why we should learn and
we have to learn Eigenvalues and
eigenvectors and I'll try to appreciate
all the audience here by motivating you
enough by the applications in science in
applied mathematics and in all the STEM
Science Technology Engineering and
Mathematics and definitely Biology as
well so second part of the presentation
is about and go into the depth of basics
of eigenvalues eigenvectors especially
complex eigenvalues and its mathematical
modeling and application of complex
eigenvalues third I will enter into the
applications of real and complex
eigenvalues in science especially in
Aerospace broadly Mechanical Sciences,
Applied Mathematics broadly Mathematical
Sciences and yes definitely
Data Science then I will touch upon the
stability and phase portraits for
especially for PhD and Post-Doctoral
students and Faculty who are interested
in nonlinear dynamics and chaos which is
one of my favorite subject and I will
end the presentation with live
demonstrations which I would like to
start and collaborating with couple of
institution like AMS and MAA
on the field of Visual Mathematics by
demonstrations of complex eigenvalues
using Mathematica all right so to begin
with as I promised so this slide is
dedicated towards
that why should I learn eigenvalues
and are they really important yes so let
me start by addressing again and raising
two questions so I'm asking again
can eigenvalues crash an airplane can
eigenvalues collapse a bridge these are
very fascinating, interesting and
mind-boggling thoughts let me show you
something which I really believe is
going to excite you also enough are you
able to see this video hello
excuse me is this video visible is this
visible I think you're mute - no ma'am
okay visible now, These are experiments performed in the NASA Langley Labs earlier to demonstrate
describe the phenomenon which is an era
aero-elastic instability phenomena known
as flutter we will see in the second
video the formal definitions of flutter
quickly but I would like to show you
this video to actually let you
appreciate the power of eigenvalues and
eigenvectors and what chaos they can do
if we really don't really make them bounded yeah
Most of these are
the Windtunnel tests of the full scale
and little bit on the scaled-down models
in the Windtunnel testing and you
will see also the full scale of models
which are experiencing aeroelastic
phenomena known as flutter which is an
n aeroelastic instability and it is all
happening let me also let you draw your
attention towards it's all happening to
because it values really getting excited
which are the natural frequencies of the
system because of the forcing frequency
from the
aerodynamic forces yeah
so here what we have seen is been actual
testing videos launched and issued by
NASA so let me also ask to excite you again on the topic which was about
to talk about that can eigenvalues
collapse a bridge so let me show you one
more very interesting demonstration of
flutter so if you are not able to see
the video please raise your hand
else I will let it play
Now, you are going to learn formal definition of flutter and also observe the importance of Eigenvalues in Science. Stay tuned...
on the tail of an aeroplane is really
vibrating like a spring elastic strain
just happening because of the
self-excited oscillation in the
aerodynamic and aeroelastic phenomena
and later in the second half of the
video we are going to experience the catastrophe, which eigenvalues or aeroelastic instability induced self-excited oscillations can have on a Bridge. Observe carefully and stay tuned...
in the earlier 1940,s so what happened was
so we were what you are seeing is the
natural frequency and especially the eigenvalues of the structural
components were matching with the
functional frequencies coming from the
forcing function from the right hand
side of the equation which we see that
so that literally led to this
catastrophe
This is the real-time footage captured, when the whole civil structure is undergoing catastrophic oscillating. Observe the structural failure and appreciate the importance of eigenvalue analysis
Now, let's understand the three segments bifurcated by the eigenvalues.
Observe the picture on the bottom right corner of this slide
Here you will observe that numerical value of eigenvalue decides,
whether the system (in this case, aeroplane or bridge) will undergo stable oscillation, or Flutter or Divergence.
Can you observe? Let's understand...
on the right-hand side of this particular slide you can see the inset of the
three particular parts of the figure
where one of them is the stable region
the second one is the flutter region the
third one is the divergence which is the
fundamental phenomena in stability again
so what we saw is the phenomena which is
happening the middle part of the
vibration phenomena in the aeroelastic
instability that is flutter so this was
the the whole idea of this particular
interaction which I generally like to
interact with people when I am in the
giving lectures but physically is that
to have more understandings around that
what is the effect and influence of the
mathematical modeling in these type of
critical systems which are catastrophic
incidents which we have already seen in
the past so we'll discuss about that you
know what and which companies are really
build on eigenvalues and eigenvectors
but the let me begin now with the basics
of eigenvalues and eigenvectors and then
we'll go into the other part of the
presentation so in this part I'm going
to talk about that what are eigenvalues
what are eigenvectors what are complex
eigenvalues the mathematical modeling
and the applications of complex
eigenvalues so let's begin
eigenvalues and eigenvectors are
instrumental to understand electrical
circuits mechanical systems ecology and
even google's page-rank algorithm we most of
us are aware about of this particular
equation which is one of the fundamental
equation in the matrix algebra and which
we call as eigenvalue problem or EVP so
eigenvectors are those vectors whose
direction remains unchanged when
transformed especially linear
transformation happens via a fixed
linear transformation say T
where eigenvalues are the values of
the extension factor or the scaling
factor so this is the
definition of eigenvalue problem but
in geometrical form to build an
intuition around this Ax is equal to
lambda x eigenvalue problem let
me show you this cartoon so over here
in a 2D system in a Cartesian plane
let's say we have a vector a 2 cross 1
vector
so with coordinate X and y represented
so the bold red line over here
represents a baseline vector X in this
equation right in the left hand
side now when it is operated by a matrix
A or it goes under a linear
transformation then what happens if the
matrix (A) leads to the vector (x) getting
scaled either scale up or scaled down
only skip no rotation right so then that
particular vector in this case X is
known as eigenvector and the
value by which it scales in this
scenario scale up is known as eigenvalue
therefore if you see the vector X
earlier which is a red bold line
represented over here when it is
multiplied by matrix X then it gets
scaled up by value lambda and then it
becomes lambda X then the lambda is
eigen value and the vector X is the
eigen vector yeah so from this equation
perspective our interest is to find only
non-trivial solution of eigen vectors X
that is non zero vectors using the
Cramer's rule which says that the
determinant of the A minus
I should be produced zero to ensure that
the mathematical sense that the matrix A
minus lambda I is singular so its
determinant should be zero this
particular equation determinant of A
minus lambda I is called as
characteristic equation and its roots
are the eigenvalues definitely
mathematicians know that by
the fundamental theorem of algebra every
polynomial of degree n has n solutions
in C then the degree of characteristic
equation will have n number of
eigenvalues associated to that system
so this little bit of mathematics
definitely all of you and believe shall
be able to appreciate but if are there
any questions I'll be happy to address later on
now let's move forward now we
have the EVP (Eigen Value Problem) now let's see that
how we can obtain the EVP the fundamental
question which always arise in the mind
of students and scientists alike is that
okay I have understood
I appreciated eigenvalues I have learned
about it but generally where do I find
them so the second thing which I'm
trying to let you appreciate is that the
the kind of eigenvalue problem we can
easily find in the nature let me give
you an example of a coupled linear
differential equation system in a
predictor-prey model so here what
happens is let's say x1 is prey and x2
is predator they both are
interdependent on each other right which
leads to these two differential
equations so that's why they are they
are coupled because x1 rate of change of
x1 is dependent on excuse me x2 and
indeed x2 growth rate is dependent
on x1 that's why they are coupled and
these they are definitely linear
differential equations because their
degree is 1 and they represents the
differential equations over here
especially order differential equations
notice that system of differential
equations can arise quite easily from
naturally occurring situations and I
have no doubt to say that most of us are
you know might be modelling using data
science and many other domains at their
home or doing this quarantine (due to COVID19) as well as during these other you know time the the
pandemic model which is the SIR one
of the famous model the Susceptible-
Infected and Recovered model is also the
coupled differential equation model
although they are not linear so note
here we can write an authority in a
differential equation and system and the
idea is to convert that coupled
differential equation over here into the
matrix form so that we can convert
from the set of algebraic equation to the new matrix which we really want to deal with
so that we can get the eigenvalues out
of it so the above equation the coupled
set we can write that into this matrix
form that is X dash is equal to a X
where A is the matrix which means we can
form by the coefficients of these
coupled differential equations now here
the eigenvalues of matrix A will be
complex we want to see that and then
what happens we know that the equation
is of the form of a pure exponential in
this scenario when we have X dash is
equal to Ax then we obtain when we put
that values into this equation we get
our eigen value problem will be started
with and so the objective as I said
earlier the the objective is to find the
determinant of A minus lambda I x is
should be equal to 0 to really find the
lambdas and from matrix algebra
perspective to let the appreciation from
a matrix algebra as well we can say that
the vector the eigen vector x is in the
null space of A minus lambda I
and the eigenvalue lambda is chosen
such that A minus lambda I has a null
space so I really don't want to detach
the intricacies and the appreciation of
the matrix algebra but at the same time
I will also would like to have given
equal weight to the whole fundamental
phenomena of Applied Mathematics and
Applied Science and their knit that's why
I also want to highlight over there that
this is really one of the key thing
which has been developed and the whole
credit goes to be the people who did the
matrix algebra evaluations earlier so to
be of use we want to ensure that the
matrix A minus lambda I is singular that
is the determinant is equal to 0 so now
here is the is the key thing where the
question which with a small anecdote
where I started that what is the
physical inference if you really
want to understand that the
eigenvalue lambda is a complex number so
now focus please, I would like to draw
your attention towards this, now focus
on the response X of T for large times T
since eigen value is a complex number so
we can represent it as a real part of
lambda and a imaginary part of lambda
let's say lambda is equal to alpha plus
iota beta then equation response or
system response will be determined as
unstable if it approaches infinity or if
we say the lambda value the real part
of lambda that is alpha is greater than
zero then it will see that if you put it
over here then the exponential will
really blows us up; it is neutrally stable
if alpha is zero and it's stable if
the
sorry I think the the real part of
lambda is less than zero then the
exponential will decay and however for
the complex part of the eigenvalue that
is beta leads to oscillations because
it is to power iota beta if I put alpha
e raised to power, alpha plus iota
beta is equal to then e raised to power iota
beta t is a representation of we can use
the Euler identity; that is e raised to power iota theta is equal
to cos theta plus iota theta (note, for small values of theta, sin theta is equal to theta) and here the decay or growth is
governed by e raise to power alpha t
let's see by an example you know
whatever we have discussed ever in in
the last two slides let's quickly
understand by in a quick example then I
would like to relate that with the
physical inference let's take a
predictor-prey model again here which
earlier we modeled in the matrix
equation that is x dash is equal to Ax
and with an initial condition that x0
it becomes the IVP initial value
problem the eigenvalues and the
eigenvectors of this matrix we can
calculate so I am also giving you the
solution so you can really do this
exercise later on but in the interest of
time I'll quickly go through this so the
determinant we find and here
because it's the 2x2 matrix will get the
polynomial which is the characteristic
equation and equate it to 0 and we'll
try to find it it's a quadratic because
a 2x2 matrix as I said earlier because
the fundamental theorem of algebra then
we find the two roots of this equation
these are the eigenvalues then the one
in lambda 2 and they are conjugate pairs
because they're complex number over here
and now the eigenvector for the first
eigenvalue let's say lambda1 is
2+ 8i we need to solve this
particular equation right so when I put
lambda over here then these are my eigen
vectors if I really put it over here
then the first eigen vector is this
now the solution corresponding for the
found eigen value and eigen vector we
can Club it the general solution
because it's a linear differential
equations as the principle of
superposition holds true here and we can
write the solution as the addition over
here because of the observed two
eigenvectors and an associated two
eigenvalues now that if we apply the
initial conditions to find these
constant c1 and c2 we can easily find
that their values from the initial
conditions from our IVP initial value
problem now then using so we started
with the equation for the predictor-prey equation which is a set of linear
equations in linear differential
equation and then we
formed an EVP eigenvalue problem then we solve that eigenvalue problem found that
eigenvalues are the complex eigenvalues
and then we found its eigenvectors
associated eigen vectors using
superposition we constructed this
particular actual solution so the
eigenvalue is written over here and here
in the left side you see this particular
plot this is known as phase portrait of
this dynamical system (predictor-prey model) so phase portrait is a representation of the
dynamical system which tells us about
the stability of the system the
dynamical response stability so here we
plot the 2 x1 and x2 responses of the
system together to see that whether
the system will be stable neutrally
stable or unstable and these as I said
earlier so this is the crux of that if
somebody asks you that what is the
physical interpretation of a complex
eigen value or rather an eigenvalue in
general then this is the the whole
summary which you should really
definitely a takeaway message that if
the real part of the eigenvalue lambda or
alpha is greater than zero then the
solution will be unstable in this
scenario this is unstable because the
solution is really blowing up it can be
neutrally stable if alpha is zero or
asymptotically stable if alpha is less
than zero and if you see the response
which we'll see that in a live
demonstration very soon that beta the imaginary part of the complex eigen
value represents the oscillations which
happens in the dynamical system so this
is the this slide is the most important
take-home slide yeah let's quickly get
into the applications of eigenvalues so
here I have listed some of the
applications of eigenvalues as you see
that the eigenvalues are used
extensively in aerospace mechanical
sciences applied mathematics civil
engineering, ECE and electrical engineering
artificial intelligence machine learning
quantum mechanics and of course this is
one of my favorite the some industries
are also built on eigenvalues and
eigenvectors and one of the biggest
industries which touches all of us
indirectly or directly is Google and it
is build on eigen vectors only
eigenvectors in late 90s when paper came
out by Sergey Brin as a research project
in Stanford let me quickly go through
that when I say that it is quite highly
used in the these particular fields
which I have listed earlier so
eigenvalues are used in designing
bridges the demonstration of that we saw
that in the motivational slide so the
natural frequency of the bridge is the
eigenvalue of smallest magnitude which
has the maximum energy area in the eigen
mode that models the bridge so engineers
especially civil engineers for designing
bridges exploit this knowledge to ensure
the stability of the constructions in
communication systems especially for ECE
students and EE you all will be able to
appreciate Shannon because of Shannon
theorem and the lot of contributions by
Shannon so eigenvalues were used by
Shannon the determinate the theoretical limit to show how much information can be
transmitted using mediums like telephone
or through the air in structural
dynamics aerodynamics or mechanical
sciences in general eigenvalues are
extensively used for structural dynamics
modal analysis dynamical stability of
structures dynamic buckling CFD
computational fluid dynamics computation
fluid dynamics and computational
structural dynamics interactions (CFD-CSD) which
is also known as FSI Fluid structure
interactions finite element analysis etc
as many of you will be already quite
equipped with the subject of machine
learning because of the courses you are
going through in your curriculum and
the industries also galloping so it is
extensively used item values are also
used in machine learning and AI one of
the key contributions is using PCA that
is principal component analysis which is
used for dimensional reduction of big
data or large datasets which also
increases the interpretability
but at the same time minimizes
information loss it does so by creating
uncorrelated variables and successively
maximizing the variance so it
essentially finds the principal
components, reduces to solve an EVP and finds
eigenvalue and the principal
eigenvectors associated with that
eigenvalues and eigenvectors are
extensively used in quantum mechanics
which is also touching and influencing
quantum computing which is also one of
the key fundamental and galloping area
of research in the today's era so the
schrodinger equation for the harmonic oscillator
have schrodinger wave functions which
are also eigen functions so let me show
you quickly the demonstrations which I
promised you earlier by visual
mathematics so what gives our show and
discussed earlier was the appreciation
of the subject of eigenvalues especially
complex eigenvalues by its applications
and the catastrophic can do if you
really don't put up due attention to it
second we saw that whatever we have how
we divide and extract the the
fundamental mathematical models from the
physical systems and then throw the
eigenvalue problem how we can solve the
eigenvalue problem get system response and
plot its phase portraits and then we saw
its applications in the standard areas
of engineering and alike now let me quickly
go to the demonstration and for that
I'll be switching my screen and to go
to the Mathematica so are you able to
see my screen yeah okay thank you so
here you can see one of the phase
portrait plots these are basically known
as flow maps also in nonlinear dynamics
or modeling and simulation courses so
over here if you see that only we
discussed about one fundamental equation
that is x dash is equal to Ax right so
over here and where my mouse is hovering
I'm actually showing you that only where
I have actually exploded a by its two by
two matrix entries a11 a12 a21 and a.22
and yes x1 and x2 are the responses of
those coupled linear differential
equations which can arise from all these
kind of subjects which you saw that in
the third part of the presentation from
any of the application area that can be
aerospace and alike right so now here this
is this is really something which I
always wanted to really show and discuss
with the students and I always wanted to
learn in this way but I learned it
later on and that was my motivation to
share it with you all so here you can
see that this is the visual part of the
mathematics which is a subject really
coming along today globally so here what
we can do is we can really manipulate
these entries rather than actually in a
MATLAB or in any kind of system
equivalent like python these days people
trying to really put these matrices
and then plot it using the libraries in
build or external but here using
Mathematica you can do that in a really
demonstrated way so if I really
increase the number of any of the entry
associated then and I actually indeed
are updating this particular equation
and what I'm updating is deliberately
I'm updating one of the entry at a time
by doing so you can see over here the
flow map changes and if I have to say so
here in these flow maps so you see
the x1 and x2 both are plotted then
how their response is changing right and
in the left hand side you can see the corresponding eigenvalues
where the horizontal line represents the Re, that is real part of the
eigenvalue remember which is alpha and
on the vertical line it represents
imaginary part represented by im
imaginary part of the eigenvalue lambda
and that was beta right you can please
refer to the presentation which shall be
shared by the organizers so as I change
the entry please observe I
this matrix I change this value I change
this matrix then the eigenvalues also
change see right this is so sensitive
that's why eigenvalues are very
critical to understand deeply if you
really mess up with the understanding of
eigenvalues and especially complex
eigenvalues it generally happens and are
observed in all the natural phenomenons
especially in buildings, if any one of
you have visited Dubai you have seen
these skyscrapers over there they
all go through rigorous testing and the demonstration of the knowledge of civil engineering is really
fantastic and they all are really masters of this
particular fundamental of complex
eigenvalues because this is what is
actually happening in the background
like if mathematics is always at the
play right let me show you one more
thing you can play with this I can share
these codes if any one of you like on my
LinkedIn or github page and then if
anybody you would like to see that and I
will share that with you let me
show you one more thing as
I promise now here you see, earlier we saw
the phase portraits which represents the
stability of the system which is
critical from a nonlinear dynamics
perspective but we live in a system that
we would like to see in the system
response with respect to time to tell me
so here is the solution so now on
the x-axis is the time on the y-axis is
y1 and y2 which are the response of the
two vectors and again if you see I have
the entries of that equation x dash is
equal to Ax where a is A matrix a 2 by 2
matrix and it's 2 by 2 entries are
written over here a11 a12 a21 and a22
and as I change please look at it
carefully as I little bit perturb any of
the values system can be stable if I
change it a little bit system can really
go into explosion you see the values
over here 10 raised to power 9 right so
similarly you can look into these
understandings yourself to see that what
is the criticality of eigenvalues and
especially complex eigenvalues and what
are its importance in our day-to-day
life
alright I'm switching to the slide ok
how much time I have I think I'm on time
it's going to be 5 so in the interest of
time so I'm closing my presentation with
these two particular questions which I
have asked to the audience
I'll give the answers to that but if
anybody of you would like to further
explore and would like to discuss with
me you're more than welcome for that
okay can you tell as I asked you the
famous algorithm which utilizes
eigenvectors and which revolutionized the World Wide Web
the answer is page rank algorithm and
the paper which was a seminal paper
which really founded Google which is our
lifeline of today and definitely which
is part of the alphabet parent company
and it came out as a research article
from Sergey Brin and Larry Page you can
look into that second question and the
last question was can you tell me the
worth of eigenvectors in 2004 for google
it was 1.9 billion dollars in 2004 when
it was listed its IPO was listed first
time so in 2004 remember I mean so
so I'll end my presentation as I promise
to respect the time it's 4:59 and I
leave you with its thought and the
takeaway message already send you the
slide and with this particular comment
make complex easy please that's the
responsibility of all engineers and
scientists alike and thank you for
giving me this opportunity and all of
you are encouraged to interact with me
and give me the
feedback to further improve and also
share your thoughts and collaborate
thank you all
not able to listen to you yeah yes please
yeah thank you very much again it was
really wonderful talk even I was lost at
your screen and so it was really a
wonderful
