Spectral Theorem for a Self-Adjoint Operator in Finite Dimensions

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

Then there are an orthonormal basis $e\in V^{n}$ of $V$ and an $n$-tuple $\lambda\in\mathbb{R}^{n}$ of real numbers such that

$$T(e_{k})=\lambda_{k}e_{k}\qquad\text{for every }k\in[n].$$

In particular each $e_{k}$ is an eigenvector of $T$ with eigenvalue $\lambda_{k}$.