Conditions under which eigenvalues of hermitian matrix are positive

Question

Under what conditions are the eigenvalues of a hermitian matrix positive?

Answer 1

For a hermitian matrix $A$, the following are equivalent:

1. $\mbox{Spectrum}(A)\subseteq (0,+\infty)$.

2. $(Ax,x)>0$ for every $x\neq 0$, i.e. $A$ is positive definite.

3. There exists $B$ invertible such that $A=B^*B$ (which also implies that $A$ is hermitian).