Inverse of non-square matrix, pinv not working....

In my physics work, I need to solve for B in the matrix equation A=B*C, where A has dimensions 90000x252 and C has dimensions 64x252, so B should have dimensions 90000x64.

My thought was that A'=C'*B' so then I could use pinv to solve pinv(C')*A'=B' so B=(pinv(C')*A')'

While I understand this would not be a unique solution the problem that I am having is that the result I get for B seems to be incorrect. When I test this by multiplying B*C, I get a matrix which (other than having the same dimensions) does not resemble A.

Any help on this matter would be greatly appreciated. Thanks

Answer 1

[Walter Roberson]

You should probably be using the mldivide ('\') operator rather than pinv() and matrix multiplication:

B = A.' \ C.' ;

[Si]

I'm not exactly sure of why this is preferred, but am at rope's end so tried it. It gives me a result which seems less like random noise, but still using this solution of B, gives B*C ~= A....

[Walter Roberson]

I had the operations the wrong way around: it should have been

B = C.' \ A.'

A shorter way of expressing that is

B = A / C

But yes, B*C will not then be the same as A.

For example, let

A = reshape(1:45,9,5);

C = reshape(sqrt(1:20),4,5);

B = A / C;

then if you compare A*pinv(C) to B you will find the differences are on the order of +/- 1E-9; the algebraic tests will all be consistent to within tolerance. But B*C will differ from A by up to 0.235 . It's an algebraic fact of life that you will not get an exact reconstruction when you work with non-square matrices.

[Si]

Ok, I had tried B = A/C but that wouldn't work because C was nearly singular,s o I got very incorrect answers. I had been able to get somewhat decent reconstructions but they weren't working for my application, so I was just hoping I could find some way to improve on them....