MPINVERT: Moore-Penrose Inverse of a Matrix

[Han]

(02-06-2015 08:40 PM)salvomic Wrote:  Hi Han,
I'm interested in this extension, almost for the case in witch the matrix is not invertible and MPINVERT doesn't work.
I saw the SVD syntax. The guide writes "Description
Singular Value Decomposition. Factors an m n matrix into two matrices and a vector: {[[m m square orthogonal]],[[n n square orthogonal]], [real]}.".
Now I try svd([[1,2][1,2]]) and I get:
a matrix [[1/√2,0][1/√2,0]] then a vector [3.16227766..., 0] and at the end another matrix [[0.447213..., -0.894427...][0.894427..., 0.4472135955]]
What it the "real"?

The vector of "real" values is a list of the singular values (i.e. the diagonal entries of \( \Sigma \).

Quote:Those values are U, ∑, V and ∑ is the diagonal (√10, 0). Is it right?

Yes. The help has been updated to correct the ordering of the output so that now it does in fact return \( \{ U, \Sigma, V \} \).

Quote:And therefore, how can we operate to get ∑, A+ and so on?
Eddie, then, could improve it's program for the case of singular non invertible matrix...

The pseudo inverse is \( V \Sigma^{-1} U^{H} \) where \( U^H \) is the Hermitian (conjugate transpose for complex matrices, or just transpose for real-valued matriced) and \( \Sigma^{-1} \) is simply the the transpose of \( \Sigma \) with the diagonal entry \( s_i \) replaced by \( \frac{1}{s_i} \). For values of \( s_i \) that are very small, (i.e. 0), then \( s_i \) is simply replaced by 0.

Quote:Another thing: in that case Prime apparently give a vector with 3 elements (matrix, vector, matrix): there is a simple way to extract one of them, i.e. to make ration approximation (with QPI)? Please, help.

You can use mat2list, and QPI accepts lists as input.

Currently, the SVD() command is not as robust. I recommend using: http://hpmuseum.org/forum/thread-4976.html

I have updated the program to also include pinv() for pseudo-inverse as well as pivoted QR factorization.

This article has a decent interpretation of the SVD (though nowhere complete, however): http://robotics.caltech.edu/~jwb/courses...pseudo.pdf