topic Re: How to test the singularity of a matrix in IML? in SAS/IML Software and Matrix Computations

How to test the singularity of a matrix in IML?

ShufeGuoding — Thu, 19 Mar 2015 06:30:20 GMT

Is there any function or algorithm to verity a matrix is nonsingular in IML? I need if in a "if then"statement.

Re: How to test the singularity of a matrix in IML?

IanWakeling — Thu, 19 Mar 2015 07:15:54 GMT

The function DET returns the determinant of a matrix, so:

if det(x)^=0 then ......

but you may want to exclude small determinants if whatever you want to do is likely to give stability problems with near-singular x, so something like:

if abs(det(x))>s

where s is some suitable small value.

Re: How to test the singularity of a matrix in IML?

Rick_SAS — Thu, 19 Mar 2015 12:23:08 GMT

The problem you describe is equivalent to computing the "rank of a matrix." A nonsingular matrix is a square matrix that has full rank. Because of finite-precision arithmetic, the rank problem is difficult. As Ian points out, there are problems with trying to compare a floating-point computation (determinant, eigenvalue, singular value,...) with zero. Stable rank-detecting algorithms use some variation of Gaussian elimination to determine whether there are linear dependencies in the rows or columns of a matrix. In SAS/IML, the standard way to detect rank is to compute the generalized inverse (which always exists) and then multiply it by the original matrix. If you get the identity matrix, then the original matrix was full rank (nonsingular). Otherwise the original matrix is less than full rank. So to detect singularity, I suggest the following:

A = { /* your matrix to test */ };

matrixRank = round(trace(ginv(A)*A));

if matrixRank < nrow(A) then print "Singular";

else print "Nonsingular";