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I have a [itex]2\times 2[/itex] real matrix [itex]M[/itex] such that: [tex]M=A^T \Sigma A[/tex], where the matrix [itex]\Sigma[/itex] issymmetric positive definite, and [itex]A[/itex] is an arbitrary 2x2 nonsingular matrix. BothAand ∑ are unknown, and I only know the entries of the matrixMitself. Note thatMis symmetric positive definite too.

I was wondering if it is possible to apply some decomposition of the matrix [itex]M[/itex] in order to find another matrix [itex]P_M[/itex] such that: [tex]P_M = AQ[/tex]

where the matrixQmust not depend onA(e.g. it cannot be a product of matrices whereAappears). I basically want to find a matrixP_{M}where the multiplication withAappears only at the left side, and not at both sides like inM.

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# Question on decomposition of a matrix

Can you offer guidance or do you also need help?

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