Product of maximal rank matrices with a positive definite matrix

Question

Consider $A_{n \times n}$ which is a positive definite matrix and $H_{m \times n}$ and $G_{n \times m}$ which are maximal rank matrices where $m \lt n$. Can we say that $H A H^T$ is positive definite? What about $H A G$?

Answer 1

First we notice that $H$ has full row rank $\Rightarrow$ $H^T$ has full column rank $\Rightarrow$ $\ker H^T=0$. Check that $HAH^T$ is pos.def. by definition:

1. $\forall x\colon$ $x^THA\underbrace{H^Tx}_{z}=z^TAz\ge 0$.
2. Let $x^THAH^Tx=z^TAz=0$ $\stackrel{A\text{ pos.def.}}{\Rightarrow}$ $z=H^Tx=0$ $\stackrel{\ker H^T=0}{\Rightarrow}$ $x=0$.

Thus, $HAH^T$ is pos.def.

P.S. For $HAG$ try $G=-H^T$. Moreover, $HAG$ is not symmetric in general.