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Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.

Created over 8 years ago, updated 10 days ago

Theorem. A matrix $A$ is Hermitian if and only if multiplication by $A$ commutes with the standard inner product on $\mathbb{C}^n$:
$$\langle Ax, y \rangle = \langle x, Ay \rangle \quad \text{for all } x, y \in \mathbb{C}^n$$

Proof.
$$\langle Ax, y \rangle = y^*(Ax) = (A^* y)^* x = \langle x, A^* y \rangle$$
So $\langle Ax, y \rangle = \langle x, Ay \rangle$ for all $x, y$ if and only if $A = A^*$.

This property says that a Hermitian operator is self-adjoint: it is its own adjoint with respect to the standard inner product. This is the fundamental property that makes Hermitian matrices central to quantum mechanics, where observables are represented by Hermitian operators.