# Wierd sum of squares compilation to a single vector norm

1. Apr 17, 2012

### Sorento7

I've encountered a function like this:
S(x) = [M(x) - F(x)] ^2 + || G(x) || ^ 2

X being a 3*1 vector
M and F: vector----->scalar
G: vector------->vector and || G || meaning its norm

To change S(x) into a single square, authors have described it like this:

S(x) = || A + B || ^ 2 where A=(M(x) - F(x) , 0) and B=(0 , G(x))

I don't understand how two vectors could be added actually resulting a vector with the first eigenvalue being scalar and the second eigenvalue a 3*1 vector itself?
i.e. what is the "nature" of A + B? a vector?

2. Apr 17, 2012

### Office_Shredder

Staff Emeritus
Why are you describing the entries of the vector as eigenvalues?

All that they are doing is this. Suppose that
$$G(x)=(G_1(x),...,G_m(x))$$
where each Gi takes a vector to a scalar. Then
$$S(x) = (M(x)-F(x))^2+G_1(x)^2+...+G_m(x)^2$$
We are adding the squares of a bunch of numbers together. This is the same as taking the norm of a vector which has those numbers as its entries
$$S(x) = ||\left( M(x)-F(x),G_1(x),....,G_m(x) \right) ||^2$$

This vector whose norm we are taking is equal to A+B in your post

3. Apr 17, 2012

### Sorento7

Re: weird sum of squares compilation to a single vector norm

Thanks this sheds some light. Forgive my rudimentary mistakes since I am actually a doctor!

Well, the problem was more complicated than I described:

A(x) = (F(x) - M(x) + ∂M/∂x * x )2 taking 3*1 vector x to scalar
B(x) = || K.x ||2 k:3*3 matrix

I am confused how can (A + B) be represented as :

|| (F(x) - M(x) , 0) + (∂M/∂x , K).x ||2

Especially, what would be the dimension of the matrix (∂M/∂x , K) ? (discrepancy between the first 1 * 3 entry and second 3 * 3?)