Spectral Theorem

The spectral theorem is a family of results in linear algebra and functional analysis describing when a linear operator can be represented by independent, orthogonal spectral components. In finite dimensions, it characterizes operators admitting an orthonormal basis of eigenvectors. In infinite-dimensional Hilbert spaces, the corresponding representation generally uses a spectral integral rather than an eigenvector expansion. The principal versions concern self-adjoint operators and, over complex spaces, normal operators. (linear.axler.net)

Finite-dimensional formulations

For a real symmetric matrix $A$, satisfying $A^{T}=A$, the theorem states that all its eigenvalues are real and that its eigenvectors can be chosen as an orthonormal basis. Equivalently,

where $Q$ is an orthogonal matrix and $D$ is diagonal with real entries. The columns of $Q$ are eigenvectors, and the corresponding diagonal entries of $D$ are their eigenvalues. This is stronger than ordinary diagonalization, which does not require an orthonormal change of basis. (math.berkeley.edu)

For complex matrices, write $A^{*}$ for the conjugate transpose. A normal operator satisfies

The complex spectral theorem states that $A$ is normal if and only if

for a unitary matrix $U$ and a complex diagonal matrix $D$. A self-adjoint operator, also called Hermitian in matrix terminology, satisfies $A=A^{*}$; its eigenvalues are real. Unitary matrices are normal too, but their eigenvalues lie on the unit circle rather than necessarily on the real axis. (linear.axler.net)

The distinction between real and complex scalars matters. A real normal matrix need not be diagonalizable over the real numbers: a planar rotation through an angle other than an integer multiple of $\pi$ has no real eigenvectors. Real self-adjointness, rather than normality alone, characterizes orthogonal diagonalizability over the real field. (linear.axler.net)

Orthogonal spectral decomposition

Grouping eigenvectors by their distinct eigenvalues gives a basis-independent form:

Here $P_{\lambda}$ is the orthogonal projection onto the eigenspace associated with $\lambda$. These projections satisfy

Thus the space splits into mutually orthogonal subspaces on each of which the operator acts simply by multiplication by a scalar. Although individual eigenvectors within a repeated eigenspace are not unique, the eigenspace and its projection are uniquely determined. (math.berkeley.edu)

A standard proof for real symmetric matrices proceeds by induction on dimension. After finding a real unit eigenvector $v$, symmetry implies that its orthogonal complement is invariant under $A$: if $w$ is orthogonal to $v$, then

The restriction to this complement is again symmetric. Repeating the argument constructs the required orthonormal eigenbasis. The inner product is therefore essential to the theorem, not merely an auxiliary choice of notation. (math.berkeley.edu)

Compact operators

A particularly direct infinite-dimensional analogue holds for compact self-adjoint operators. On a separable Hilbert space, such an operator has an orthonormal basis of eigenvectors, including a basis for its kernel. Every nonzero spectral value is an eigenvalue of finite multiplicity, and zero is the only possible accumulation point of the nonzero eigenvalues. (ocw.mit.edu)

Using normalized eigenvectors $e_j$, the action can be written

with the inner product convention chosen to be linear in its first argument. The series converges in Hilbert-space norm. Compactness makes this decomposition resemble a finite-dimensional diagonal matrix, but zero can belong to the spectrum without being an eigenvalue—for example, when infinitely many nonzero eigenvalues approach zero and the kernel is trivial. (ocw.mit.edu)

Spectral measures and continuous spectrum

General self-adjoint operators need not possess any eigenvectors. Their spectrum can include continuous components, so an ordinary sum of eigenprojections is insufficient. The general theorem uses a projection-valued measure $E$, assigning orthogonal projections to Borel subsets of the real line, with countable additivity understood in the strong operator sense:

For bounded normal operators, an analogous representation integrates the complex coordinate over the complex spectrum. (pi.math.cornell.edu)

The role of measure theory is visible in a simple example. On $L^{2}([0,1])$, let

This bounded self-adjoint operator has spectrum $[0,1]$, but no nonzero eigenfunctions: an eigenfunction for $\lambda$ would have to vanish except at the single point $x=\lambda$, a set of Lebesgue measure zero. Its spectral projections are multiplication by indicator functions,

Spectral decomposition therefore does not generally mean decomposition into actual eigenvectors. (math.mit.edu)

For an unbounded self-adjoint operator, the spectral integral also determines its domain. Setting $\mu_x(B)=\langle E(B)x,x\rangle$,

Self-adjointness includes equality of the operator and adjoint domains; symmetry alone does not suffice for this theorem. (pi.math.cornell.edu)

Functional calculus and applications

The spectral representation defines a functional calculus:

In finite dimensions this becomes $\sum_\lambda f(\lambda)P_\lambda$. It provides a common framework for operator powers, exponentials, and square roots of positive operators. (math.mit.edu)

In principal component analysis, a covariance matrix is symmetric and positive semidefinite. Its orthogonal eigenvectors give principal directions, while its eigenvalues give the associated variances. In quantum mechanics, self-adjoint observables have spectral measures; the Born rule assigns probability $\langle E(B)\psi,\psi\rangle$ to an outcome in $B$ for a normalized state $\psi$. (theory.stanford.edu)

The related singular value decomposition applies to arbitrary rectangular matrices. Unlike spectral diagonalization, it generally uses different orthonormal bases in the domain and range, and its singular values are nonnegative rather than arbitrary eigenvalues. (axler.net)