Question on decomposition of a matrix

[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 P_M where the multiplication with A appears only at the left side, and not at both sides like in M.

 

[fresh_42]

The Cholesky normal form for symmetric positive definite matrices $M$ gives us $M=LDL^\tau$ as yours. But we have additionally that $D$ is diagonal and $L$ a lower triangular matrix. If we take $D^{1/2}\cdot D^{1/2}=D$ the square root of $D$ and set $S:=LD^{1/2}$ we can even write $M=SS^\tau$ with a lower triangular matrix $S$. For $2\times 2$ matrices, this should give you some nice conditions.