Spectral Theorem for a Self-Adjoint Operator in Finite Dimensions

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, and suppose that $V$ is \reftext{def:finite-dimensional-vector-space-2026b}{finite-dimensional} and $V\ne\{0_{V}\}$; write $n=\dim V$ for its \reftext{def:dimension-inner-product-space-2026a}{dimension} and let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$. Let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$ that is \reftext{def:self-adjoint-operator-2026b}{self-adjoint}.

Then there are an \reftext{def:orthonormal-basis-2026b}{orthonormal basis} $e\in V^{n}$ of $V$ and an \reftext{def:finite-tuple-power-2026a}{$n$-tuple} $\lambda\in\mathbb{R}^{n}$ of \reftext{def:real-numbers-c54-2026c}{real numbers} such that

In particular each $e_{k}$ is an \reftext{def:eigenvector-eigenvalue-2026a}{eigenvector of $T$ with eigenvalue $\lambda_{k}$}.