# Condition for existence of polar decomposition for a real matrix?

1. Sep 17, 2012

### Bavid

I have a matrix A with real values that I wish to decompose into a proper orthogonal matrix R and a symmetric matrix U.
Now the matrix A is obtained from a process such that this decomposition is usually possible, and U turns out to be also positive definite.

The problem arises when I get a matrix that is "close" to an identity matrix, i.e, its components differ from the identity matrix by random values on the order 1e-3.
Here, sometimes I get non-numeric (NaN) values when I attempt to do the decomposition. I can disregard these 'bad' matrices altogether, but I do not know how to check for such a 'bad' matrix.
Is there a condition on A such that A=RU is possible for real-valued matrices? I can guess this question reduces to the existence of matrix square root for ATA, but not sure of how to proceed further...