Date: Nov 16, 2011 2:27 AM
Author: Noob
Subject: How to find all SPD matrices bounded by a specific SPD matrix?

Assuming that A and B are symmetric positive definite(SPD) matrices, and we say A>B if A-B is a symmetric positive definite matrix.
Question: find all symmetric positive definite matrices C such that C<A.
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Comment: I had a feeling that a certain property was satisfied by all SPD matrices that are "less than" A. However, when I tested my conjecture in Matlab, I didn't know how to test all such matrices C. The way I used to generate C was that: I generated a random n by n matrix r by the command "randn(n)" in Matlab, then let C=r*r' if C<A, or re-generate r otherwise, because r*r' is a SPD matrix if r is invertible("r'" means the transpose of r). Nevertheless, it's almost impossible to generate a suitable r randomly if the dimension n is large, so I am looking for an algorithm that can exhaust all matrix C that is less than A. I don't know if there has been some research on this problem.
Any idea suggested will be appreciated!
George