(Matrix Optimization) Optimization method for coefficient matrix in Ax = b with known x and b.

Hi,

I have two 4*1 data vectors x and b which represents meaured 'Intensity vector' and 'Stokes vector'. These two vectors are related to each other by a 4*4 transfer matrix A as Ax = b. In the ideal case, the relationship for all x and b must satisfy Ax = b for an ideal transfer matrix A_ideal = [0.5 0.5 0 0;0.5 0 0.5 0;0.5 -0.5 0 0;0.5 0 -0.5 0].

Now, in a non-ideal system I want to predict the matrix A for some x and b which will satisfy Ax = b. In the optimization routine A can be initialized as A_ideal and will optimize after each iteration until Ax = b is satified (or we get an near approximation).

What is the best method to optimization the above matrix and how can I implement it?

Thank you!

Answer 1

[Torsten]

If x1, b1, x2, b2 are your given data, use

x1 = [2 7 56 -3];
b1 = [12 -4 0.23 pi];
x2 = [-5 23 9.5 12];
b2 = [0.43 2.4 -12.67 exp(2)];
n = 4;
M = [x1,zeros(1,n),zeros(1,n),zeros(1,n);...
     zeros(1,n),x1,zeros(1,n),zeros(1,n);...
     zeros(1,n),zeros(1,n),x1,zeros(1,n);...
     zeros(1,n),zeros(1,n),zeros(1,n),x1;...
     x2,zeros(1,n),zeros(1,n),zeros(1,n);...
     zeros(1,n),x2,zeros(1,n),zeros(1,n);...
     zeros(1,n),zeros(1,n),x2,zeros(1,n);...
     zeros(1,n),zeros(1,n),zeros(1,n),x2];
R = [b1.';b2.'];
A = lsqlin(M,R);
norm(M*A-R)

ans = 4.1460e-15

A = reshape(A,n,n).'

A = 4×4

0 -0.0736 0.2235 0 0 0.1411 -0.0891 0 0 -0.5826 0.0769 0 0 0.3143 0.0168 0

A*x1.'-b1.'

ans = 4×1

1.0e-14 * 0.3553 -0.0888 0.1305 0.0888

A*x2.'-b2.'

ans = 4×1

1.0e-14 * 0.0833 0.0444 -0.1776 0

[Nipun Vashistha]

Thank you so much for the help!

Answer 2

[Torsten]

If you have two 4x1 data vectors x and b to estimate A, you have 8 equations in 16 unknowns.

This system is underdetermined - you can imagine that in general, there are infinitely many ways how A can be chosen.

Is this ok for you ? Or are there some restrictions on the elments of A that should be incorporated ?

[Nipun Vashistha]

Yes, I have 4 equations and 16 unknown parameters of matrix A for each x and b. It is okay for me find any A that satisfies Ax = b. There are no restrictions on the element of matrix A.

Could you please let me know any iterative method that can applied to this problem to find an optimized A?