Positive definite and hermitian

Question

Let $H$ be quaternion space. Let $A\in M_{n}(H)$. Then $A$ is hermitian i.e, $A^{*}=A$ if and only if $A$ is positive-definite i.e, $x^{*}Ax$ is real and positive for every nonzero $x\in H^{n}$. How can I prove it?

Answer 1

Let $A$ be a Hermitian matrix. The eigenvalues of a Hermitian matrix are always positive. See if you can proceed here in the forward direction. For the other direction, let $x^*Ax$ be positive definite. Think about how you can show that the $x^*A^*x$ is equal to $x^*Ax$, given that the solution is positive definite.