Eigenvalues of a Hermitian matrix are all real. Eigenvectors $x_1,x_2$ of two eigenvalues $\lambda_1, \lambda_2$ of a Hermitian matrix are orthogonal with respect to the conjugate product. The conjugate product is defined as $x_1^Hx_2 = \bar{x_1}^Tx_2$. A proof is given as the following,
However, I am wondering if $x_1,x_2$ are orthogonal with respect to any inner product. A complex inner product is defined as the following, where $C$ is a field of complex numbers, and $F$ is the field of real numbers.
If $<x_1,x_2> = 0$ then we say $x_1,x_2$ are orthogonal with respect to $<>$. Again my question is that does the orthogonality of two eigenvectors $x_1,x_2$ holds with respect to any inner product? If not, can anyone help provide a counter example?