Symmetrix matrix multiplication and its output

Question

I have a symmetric matrix M, and I would like to multiplicate this matrix by H in the following way:

R = $HM$$H^-$

In the following case H is also a transpose matrix so:

R = $HM$$H^T$

I wonder if in any of the cases R would be also symmetric. If the answer is no for both cases, is there any exception where R would be symmetric?

Thank you for your help

Answer 1

I don't know what you mean by $H^-$. But the answer for the second case is yes:

Being symmetric means $M^T=M$. So we can simply check $R$: $$R^T = (H^T)^TM^TH^T=HMH^T=R$$ So yes, that is symmetric. The general rule for transposition of products of matrices is $$ (ABC)^T=C^TB^TA^T$$