# Question on decomposition of a matrix

1. Apr 8, 2014

### mnb96

Hello,

I have a $2\times 2$ real matrix $M$ such that: $$M=A^T \Sigma A$$, where the matrix $\Sigma$ is symmetric positive definite, and $A$ is an arbitrary 2x2 nonsingular matrix. Both A and ∑ are unknown, and I only know the entries of the matrix M itself. Note that M is symmetric positive definite too.

I was wondering if it is possible to apply some decomposition of the matrix $M$ in order to find another matrix $P_M$ such that: $$P_M = AQ$$
where the matrix Q must not depend on A (e.g. it cannot be a product of matrices where A appears). I basically want to find a matrix PM where the multiplication with A appears only at the left side, and not at both sides like in M.

Last edited: Apr 8, 2014