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.
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