Obsidian | Level 7

## How to test the singularity of a matrix in IML?

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

2 REPLIES 2
Barite | Level 11

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

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.

SAS Super FREQ

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

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";

From The DO Loop