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

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


${H}$ 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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.