CODEWITHSUNDEEP
cmatrixlinear-algebragsl
Marouen
Marouen

Reputation: 945

Computing null space of matrix via GSL with M<N gives an error

I am computing the null space of a matrix A(m,n) with gsl library in c.

Following the documentation of singular value decomposition (SVD) function in gsl such as A = USVt

" For a rank-deficient matrix, the null space of A is given by the columns of V corresponding to the zero singular values. "

The corresponding function is

int gsl_linalg_SV_decomp(gsl_matrix * A, gsl_matrix * V, gsl_vector * S, gsl_vector * work)

the syntax is the following:

#include <gsl/gsl_linalg.h>
#include <gsl/gsl_blas.h>
#include <stdio.h>
#include <stdlib.h>
...

int main(int argc, char **argv){
    ...
    gsl_linalg_SV_decomp(A,V,S,work);

return 0;
}

I get the following error

gsl: svd.c:60: ERROR: svd of MxN matrix, M<N, is not implemented
Default GSL error handler invoked

Indeed A has M < N.

Are you aware of another library that would compute the Null space of a matrix of M < N ? Is there a workaround with gsl?

Upvotes: 0

Views: 557

Answers (1)

dmuir
dmuir

Reputation: 4431

You can do this by computing the SVD of A' (A transpose) rather than A. The null space of A is spanned by the left singular vectors corresponding to zero singular values.

For any matrix B, just as the right singular vectors corresponding to zero singular values span the null-space of B, the left singular vectors corresponding to non-zero singular vectors span the range of B.

So if we do an SVD of A', with

A' = U*S*V'

The range of A' is spanned by the columns of U corresponding to non-zero singular vectors. But the null-space of A is the set of vectors perpendicular to the range of A'. Therefore the null-space of A is spanned by the columns of U correspeonding to zero singular values.

Upvotes: 1

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