# Are the eigenvectors of different eigenvalues of an Herimtian matrix orthogonal wrt. any inner product?

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?

• The notion of Hermitian is (or can be) defined in terms of the inner product, so when you use a different inner product, all bets are off. Sep 26 '16 at 15:58

• Thank you! Could you help with a counterexample for vectors of $C^n$? I find another inner product $\text{Re} {x^Hy}$ but any vectors orthogonal wrt. the conjugate product will also be orthogonal wrt. $\text{Re} {x^Hy}$. Sep 26 '16 at 18:42
• That's not an inner product: the Re spoils linearity. Try inner product $\langle x,y \rangle = x^H Q y$ where $Q$ is a positive definite (Hermitian) matrix that does not commute with $A$. Sep 26 '16 at 20:30