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Suppose M is a n × n real or complex matrix, and we are interested in the following matrix factorization:

${\displaystyle {{M} = {H} {\Sigma }} {H} ^{*} }$

Where:

${H}$ is a n × n matrix , and ${H}$ * is the conjugate transpose of ${H}$.

${\Sigma }$ can be any n × n matrix

Question 1: What kind of matrix factorization is this ? How to find ${H}$ and ${\Sigma }$ for a given ${H}$ ? We know that such factorization will not be unique. We are just interested in an algorithm to find general solutions.

Question 2: If we restrict ${\Sigma }$ to be a positive semidefinite matrix, then what kind of matrix factorization is this ? and how to find such factorization ? Will such factorization be unique or not ? (We assume it is NOT).

Thank you in advance.

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Q1: If $H$ is invertible (a coordinate transform) then this formular gives the transformation of matrices associated to bilinear forms.

Given $M$, setting $H=I$ and $\Sigma=M$ is valid.

Q2: If $\Sigma$ is Hermitian and positive definite then $M$ is positive semidefinite at least (positive definite iff $H$ is invertible).

Given such a factorization $M=H\Sigma H^*$ and a unitary operator $U$, then $M = (HU)(U^*\Sigma U)(HU)^*$ is a new factorization.

If you restrict $M$ to be symmetric, $H$ to be orthogonal matrices, then a normal form associated to this transformation is given by https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

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  • $\begingroup$ Thank you for the reply ! $\endgroup$ – david Feb 5 at 18:37

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