# What kind of matrix factorization is this ? and how to find such factorization?

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).

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