Operators Diagonal in an Orthonormal Basis

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}. Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $e\in V^{n}$ be an \reftext{def:finite-tuple-power-2026a}{$n$-tuple} in $V$ that is an \reftext{def:orthonormal-basis-2026b}{orthonormal basis} of $V$, with components $e_{k}$. Let $\mu\in\mathbb{C}^{n}$ be an $n$-tuple with components $\mu_{k}$ in the field $\mathbb{C}$ of \reftext{def:complex-numbers-2026a}{complex numbers}.

Sums of vectors are \reftext{def:finite-sum-vector-space-2026a}{finite sums in $V$}. Let $R$ be the map from $V$ to $V$ given by

Then the following hold.

\textbf{1. (Linearity and action on the basis)} $R$ is a \reftext{def:linear-operator-2026a}{linear operator} on $V$, and $R(e_{j})=\mu_{j}e_{j}$ for every $j\in[n]$.

\textbf{2. (Self-adjointness)} If $\mu_{k}$ is a \reftext{def:real-numbers-c54-2026c}{real number} for every $k\in[n]$, then $R$ is \reftext{def:self-adjoint-operator-2026b}{self-adjoint}.