The Exercise

Let $M$ be a Hermitian operator on the state space $V$ of a quantum system. The exercise asks for the spectral decomposition of $M$: $V$ has an orthonormal basis $|1\rangle,\dots,|d\rangle$ of eigenvectors of $M$, where $d$ is the dimension of $V$, and the eigenvalues $\lambda_1,\dots,\lambda_d$ are real, so that $M = \sum_i \lambda_i |i\rangle\langle i|$ is diagonal in that basis.

Hermitian Operators

The space $V$ is any finite-dimensional inner product space, complex as in quantum mechanics or real, with inner product $(\cdot,\cdot)$.

The adjoint $M^\dagger$ of an operator $M$ on $V$ is the operator on $V$ with $$\bigl(|v\rangle, M|w\rangle\bigr) = \bigl(M^\dagger|v\rangle, |w\rangle\bigr)$$(1) for all vectors $|v\rangle$ and $|w\rangle$ in $V$. A Hermitian operator $M$ on $V$, also called self-adjoint, is an operator equal to its adjoint, $M^\dagger = M$.

An orthonormal basis of $V$ is a basis $|1\rangle,\dots,|d\rangle$ of unit vectors orthogonal to one another, $$\bigl(|i\rangle, |j\rangle\bigr) = \delta_{ij}.$$(2)

The Claim

The library states the decomposition as an equivalence.

Its forward direction is the exercise.

What Is Checked

The library proves the equivalence of the theorem for every such space and every such operator.