Let $A \in \mathbb C^{n \times n}$ be such that $<x,y> = y^{\theta} Ax$ is an inner product on $\mathbb C^n$. Prove that A is a hermitian matrix with positive diagonal entries.
The first problem I am facing is to verify the properties of the inner product space defined above. Also, I need some help regarding how to prove A is a hermitian matrix with positive entries on its diagonal.
$\langle x,x \rangle \geq 0$ for all $x$ and this implies that $A$ is a non-negative definite matrix. All non-negative definite matrices have the stated properties.