Consider a Hermitian $(n \times n)$-matrix $A$, and a Hilbert space $\mathbb{C}^n$, foreseen with a nonstandard inner product. (An inner product $s(\cdot,\cdot)$ is standard if for any two vectors $x = (x_1,\ldots,x_n)$ and $y = (y_1,\ldots,y_n)$ we have that $s(x,y) = x_1y_1 + \cdots + x_ny_n$.)
I understand that it is important in Quantum Theory that $A$ has an orthogonal base of eigenvectors, and relative to the standard inproduct, this is indeed the case.
But what about a nonstandard inproduct ? Is this property also true, or does one only work with standard inproducts in Quantum Theory ?