When does a matrix have only positive eigenvalues?

Question

When does a matrix has only positive eigenvalues?

I know that you can say if the eigenvalues are real or not by saying if the matrix is selfadjoint or skewadjoint, but how can you prove that has only positive eigenvalues?

Problem:

Let $A$ ∈ $\Bbb C^{n×n}$. Show spec($A$*$A$) ⊂ $[0, ∞).$

Answer 1

If $A^\ast A v=\lambda v,\,v\ne 0$ then $\lambda =\frac{v^\ast A^\ast Av}{v^\ast v}$ is a ratio of squared lengths, i.e. $\lambda\ge 0$. As for the question of when matrics are positive-definite, there are several tests.