Positive Definite Matrices and eigenvalues

Question

Problem:

Let $A$ ∈ ${C}^{n×n}$ be, such that for every x ∈ $C^n$ <$A$x$, x$> ≥ 0

Show that all eigenvalues of $A$ are positive or zero

I suppose that from the standart inner product in the problem we can say that $A$ is a positive definite matrix and therefore follows that the eigenvalues of $A$ are positive.

However I am not sure if that is right,could someone give a hint?

Answer 1

It's a straightforward computation. If $\lambda$ is an eigenvalue of $A$, choose an eigenvector $v$ with $\langle v,v\rangle=1$. Then $$ \lambda=\langle \lambda v,v\rangle=\langle Av,v\rangle\geq0. $$

Answer 2

Using the definition that you wrote, pick your $x = v$, where $v$ is any eigenvector, then $$<Av,v> = \lambda \geq 0$$ This tells you that no matter which eigenvector you pick to be your $x$, you will get that its corresponding eigenvalue is non-negative.