I am trying to prove that an $n \times n$ matrix is positive definite iff all of its eigenvalues are positive.
I know that if $\lambda$ is an eigenvalue then:
$Ax = \lambda x$ for eigenvalues lambda. From there I can go to:
$x^TAx = x^T\lambda x$
Is there some special property of transpose matrices I can use to just get $\lambda$ on the right side of the equation? Because then the proof would pretty much be done.