A.14. Symmetric Square Root of a Symmetric Nonnegative Definite Matrix

For a symmetric nonnegative definite matrix A, a symmetric square root denoted by A^1/2 can be obtained by using the ROOT function and the subroutine EIGEN. Specifically, since A is symmetric, we must have A = PAP' = (PA^1/2P')(PA^1/2P') = A^1/2 A^1/2 where P is orthogonal. The diagonal matrix A contains the eigenvalues of A in the diagonal places, which are nonnegative since the matrix A is nonnegative definite. Thus, A^1/2 is just a diagonal matrix with diagonal elements as the nonnegative square roots of the corresponding elements of A. Accordingly, we define A^1/2 as A^1/2 = PA^1/2P'. Thus A^1/2 is also symmetric. However, it may not be unique.

The needed SAS statements to accomplish this task are

proc iml;
a = {
10 3 9,
3 40 8,
9 8 15;
call eigen(d,p,a);
lam_half = root(diag(d));
a_half = p*lam_half*p`;
print a, p, lam_half;
print a_half ;

The symmetric square root matrix A^1/2 in the above code is denoted by A_HALF. It may be pointed out that A^−1/2 may be computed by taking the inverse of A^1/2 or by directly computing the symmetric square root of A⁻¹ instead of A using the above code.