TITLE: transpose of positive matrix is positive
QUESTION [6 upvotes]: how to prove it?
I am talking about matrixes which satisfy:
$$( Ax , x ) > 0\quad \text{ for any}\quad \;x \neq 0.$$
How to prove that $A^T\;$ is also positive?
$$x^T A  x = ( x^T  A  x )^T$$
and what?

REPLY [8 votes]: Hint: Write the inner product as $x^T A x$ and use the fact that that expression is its own transpose (since it's a 1-by-1 matrix).