Unitary operator

Last updated January 27, 2022

Definition

Let $V$ be an inner product space.

Definition. An operator $A\in\text{End}(V)$ is called unitary if $$\langle Ax,Ay\rangle=\langle x,y\rangle,\quad\forall x,y\in V.$$

Properties

Let $A$ be a unitary operator.

1. If $U\subset V$ is invariant wrt $A$, then so is $U^\perp$.
2. There exists an orthonormal basis $e_1,\dots,e_n$ of $V$ and $\lambda_1,\dots,\lambda_n$ such that $Ae_i=\lambda_ie_i$. Moreover, $|\lambda_i|=1$ for all $i$.