The Spectral Theorem

The spectral theorem is the centerpiece of this module. It says that a Hermitian operator is, in the right basis, nothing more complicated than a list of real numbers along a diagonal — and it tells you exactly how to read those numbers off as projectors. Almost every computation you do with an observable, from expectation values to time evolution, is an application of this one theorem.

Statement

Spectral theorem (Hermitian case). Let $A$ be a Hermitian operator on a finite-dimensional complex Hilbert space. Then:

All eigenvalues $\lambda_k$ of $A$ are real.
There exists an orthonormal basis of the whole space consisting entirely of eigenvectors of $A$ .

Equivalently, $A$ is diagonalizable by a unitary: there is a unitary $U$ whose columns are the orthonormal eigenvectors such that

U^\dagger A\, U = \operatorname{diag}(\lambda_1, \dots, \lambda_n).

The collection of eigenvalues $\{\lambda_k\}$ is called the spectrum of $A$ — hence "spectral."

The spectral decomposition

The most useful form of the theorem expands $A$ directly in terms of its eigenvectors. Let $\{|k\rangle\}$ be an orthonormal eigenbasis with $A|k\rangle = \lambda_k|k\rangle$ . Then

A = \sum_k \lambda_k\, |k\rangle\langle k|.

Each term $|k\rangle\langle k|$ is the projector onto the $k$ -th eigenvector (built from the outer product you met earlier). The operator is a weighted sum of projectors, with the eigenvalues as weights. To check the formula, apply it to an eigenvector $|j\rangle$ :

A|j\rangle = \sum_k \lambda_k |k\rangle\langle k|j\rangle = \sum_k \lambda_k |k\rangle\,\delta_{kj} = \lambda_j |j\rangle,

recovering the eigenvalue equation. Setting all $\lambda_k = 1$ gives the completeness relation $\sum_k |k\rangle\langle k| = \mathbb{1}$ : the eigenprojectors resolve the identity.

Grouping by eigenvalue: projection-valued form

If an eigenvalue $\lambda$ is degenerate (shared by several eigenvectors), collect all of them into a single projector $P_\lambda = \sum_{k:\,\lambda_k = \lambda} |k\rangle\langle k|$ onto that whole eigenspace. The decomposition becomes a sum over distinct eigenvalues:

A = \sum_{\lambda} \lambda\, P_\lambda, \qquad \sum_\lambda P_\lambda = \mathbb{1}, \qquad P_\lambda P_{\lambda'} = \delta_{\lambda\lambda'}\,P_\lambda.

The projectors are orthogonal ( $P_\lambda P_{\lambda'} = 0$ for $\lambda \neq \lambda'$ ), each is Hermitian and idempotent ( $P_\lambda^2 = P_\lambda$ ), and they sum to the identity. This basis-independent form is the launchpad for the next two lessons.

Beyond Hermitian: normal operators

The theorem extends to the broader class of normal operators — those satisfying $A^\dagger A = A A^\dagger$ . A normal operator is unitarily diagonalizable, with an orthonormal eigenbasis, but its eigenvalues may be complex. Hermitian operators (real eigenvalues) and unitary operators (eigenvalues on the unit circle) are both special cases. This is why both observables and gates can be analyzed with the same diagonalization machinery: they are all normal.

Why it matters

The spectral decomposition $A = \sum_k \lambda_k |k\rangle\langle k|$ converts hard operator problems into easy number problems. Want the expectation value? It is $\sum_k \lambda_k |\langle k|\psi\rangle|^2$ — an average of eigenvalues weighted by probabilities. Want a function of the operator, like $e^{-iAt}$ ? Apply the function to each eigenvalue. Want to know the measurement statistics? Read them off the projectors. The next three lessons turn each of these uses into a concrete skill.