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Spectral decomposition

Spectral decomposition is a fundamental theorem in linear algebra that provides a way to factorize certain matrices using their eigenvalues and eigenvectors, expressing a normal matrix A over the complex numbers as A = U \Lambda U^*, where U is a unitary matrix whose columns are the eigenvectors of A, and \Lambda is a diagonal matrix containing the corresponding eigenvalues on its diagonal.^[1] This decomposition generalizes the diagonalization process and is guaranteed for normal matrices (those satisfying A A^* = A^* A), which include symmetric matrices over the reals and Hermitian matrices over the complexes, ensuring real eigenvalues for the latter cases and orthonormal bases of eigenvectors.^[2] The theorem, often attributed to the broader spectral theorem, highlights the role of eigenvalues in capturing the matrix's scaling properties along eigenvector directions.^[3] In the real symmetric case, the decomposition simplifies to A = Q \Lambda Q^T, where Q is an orthogonal matrix, allowing the matrix to be reconstructed as a sum A = \sum_{i=1}^n \lambda_i \mathbf{q}_i \mathbf{q}_i^T, with each term representing a rank-one projection scaled by an eigenvalue.^[2] This form reveals key properties, such as the reality of eigenvalues and the orthogonality of eigenvectors, which form a complete basis for the vector space.^[3] For matrices with repeated eigenvalues, the decomposition remains valid but may involve eigenspaces rather than individual vectors, ensuring uniqueness up to the ordering of eigenvalues and choice of orthonormal bases within degenerate subspaces.^[1] The spectral decomposition has profound implications across mathematics and applications, enabling efficient computation of matrix powers (via A^k = U \Lambda^k U^*), solutions to differential equations, and dimensionality reduction techniques like principal component analysis (PCA), where it identifies variance directions in data.^[2] It also underpins quantum mechanics by diagonalizing observables as self-adjoint operators and facilitates optimization problems through the Rayleigh quotient, which extremizes quadratic forms.^[4] Beyond finite dimensions, the concept extends to functional analysis via the spectral theorem for bounded self-adjoint operators on Hilbert spaces, influencing fields like signal processing and partial differential equations.^[5]

Prerequisites and Overview

Definition and motivation

Spectral decomposition arises from the need to simplify the analysis of linear transformations, particularly in solving systems of differential equations and understanding physical systems in quantum mechanics. In the early 19th century, Joseph Fourier's investigation of heat conduction led to the representation of solutions to the heat equation as sums of terms involving exponential decay and trigonometric functions, effectively decomposing the problem into independent modes associated with eigenvalues of the underlying spatial operator.^[6] This approach highlighted how diagonalizing operators could separate variables and facilitate explicit solutions to partial differential equations. Similarly, in quantum mechanics, the spectral theorem for self-adjoint operators enables the representation of observables like the Hamiltonian in a basis of eigenvectors, allowing time evolution and expectation values to be computed straightforwardly through diagonal forms.^[7] Formally, the spectral decomposition of a self-adjoint operator A on a Hilbert space \mathcal{H} expresses A in terms of its spectrum via a projection-valued measure E:

A = \int_{\sigma(A)} \lambda \, dE(\lambda),

where \sigma(A) denotes the spectrum of A, a subset of the real numbers.^[8] This integral representation generalizes the finite-dimensional eigendecomposition, where A = \sum \lambda_i P_i with P_i as orthogonal projections onto eigenspaces. The primary motivation for spectral decomposition lies in its facilitation of functional calculus, permitting the definition of functions of the operator such as f(A) = \int f(\lambda) \, dE(\lambda) for suitable f. For instance, this yields A^n = \int \lambda^n \, dE(\lambda), which simplifies computations of powers and exponentials essential for dynamical systems and quantum evolution equations.^[7] In the finite-dimensional setting, the concept is illustrated simply by a diagonal matrix, such as D = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix} on \mathbb{R}^2, whose spectral decomposition is D = \lambda_1 |e_1\rangle\langle e_1| + \lambda_2 |e_2\rangle\langle e_2|, with |e_i\rangle\langle e_i| as rank-one projections onto the standard basis vectors. This trivial case underscores how the decomposition aligns the operator with its spectral properties, enabling efficient matrix operations like exponentiation.

Key concepts: Operators and spectra

Linear operators are mappings between vector spaces that preserve addition and scalar multiplication. In the finite-dimensional setting, a linear operator A: V \to V acts on a complex vector space V of finite dimension n, which can be identified with \mathbb{C}^n equipped with the standard Euclidean inner product \langle x, y \rangle = x^* y, where x^* denotes the conjugate transpose.^[9] Such operators are represented by n \times n complex matrices and are always bounded, as all linear maps between finite-dimensional normed spaces are continuous.^[9] In the context of inner product spaces, which are complex vector spaces equipped with a sesquilinear inner product \langle \cdot, \cdot \rangle satisfying \langle x, y \rangle = \overline{\langle y, x \rangle}, linear operators extend the finite-dimensional case but may not induce a complete norm unless the space is a Hilbert space. A Hilbert space H is a complete inner product space, meaning every Cauchy sequence converges, and linear operators T: H \to H are often studied in their bounded form, where \|T\| = \sup_{\|x\| \leq 1} \|Tx\| < \infty.^[10] Bounded linear operators on Hilbert spaces form a Banach algebra under composition and the operator norm.^[10] The spectrum of a bounded linear operator T on a complex Banach space (including Hilbert spaces) is the set \sigma(T) = \{\lambda \in \mathbb{C} \mid T - \lambda I \text{ is not invertible}\}, where I is the identity operator.^[11] The spectrum is a nonempty compact subset of \mathbb{C}. It decomposes into three disjoint parts: the point spectrum \sigma_p(T), consisting of eigenvalues \lambda where T - \lambda I is not injective (i.e., \ker(T - \lambda I) \neq \{0\}); the continuous spectrum \sigma_c(T), where T - \lambda I is injective with dense but not closed range; and the residual spectrum \sigma_r(T), where T - \lambda I is injective but the range is not dense.^[11] In finite dimensions, the spectrum reduces to the point spectrum, as the residual and continuous spectra are empty for matrices.^[11] The resolvent set \rho(T) is the complement \mathbb{C} \setminus \sigma(T), comprising all \lambda for which T - \lambda I is bijective with bounded inverse R(\lambda, T) = (T - \lambda I)^{-1}, an analytic function outside the spectrum.^[11] The spectral radius r(T) = \sup\{|\lambda| : \lambda \in \sigma(T)\} satisfies r(T) = \lim_{n \to \infty} \|T^n\|^{1/n}, where the limit exists and equals \inf_{n \geq 1} \|T^n\|^{1/n}.^[12] For normal operators, r(T) = \|T\|.^[12] The adjoint of a bounded linear operator A: H \to H on a Hilbert space is the unique bounded operator A^*: H \to H satisfying \langle Ax, y \rangle = \langle x, A^* y \rangle for all x, y \in H.^[10] This follows from the Riesz representation theorem, ensuring A^* exists and \|A^*\| = \|A\|.^[13] An operator is self-adjoint if A = A^*, equivalently \langle Ax, y \rangle = \langle x, Ay \rangle for all x, y \in H, implying the spectrum lies on the real line.^[10] A normal operator satisfies AA^* = A^*A, generalizing self-adjoint and unitary operators (where A^* = A^{-1}).^[13] In finite dimensions, normal matrices diagonalize unitarily.^[13]

Finite-dimensional case

Hermitian matrices

A Hermitian matrix is a complex square matrix that equals its own conjugate transpose, satisfying H = H^*. In the finite-dimensional setting, the spectral theorem asserts that every such matrix admits a spectral decomposition: there exists a unitary matrix U whose columns form an orthonormal basis of eigenvectors and a real diagonal matrix D containing the eigenvalues of H on its diagonal, such that H = U D U^*. This decomposition diagonalizes H and highlights its real spectrum and orthogonal eigenspaces.^[14] To establish this result, first note that all eigenvalues of H are real. If H x = \lambda x for a unit eigenvector x (i.e., \|x\| = 1), then \lambda = \langle H x, x \rangle. Since H is Hermitian, \langle H x, x \rangle = \langle x, H^* x \rangle = \langle x, H x \rangle = \overline{\langle H x, x \rangle}, implying \lambda = \overline{\lambda} and thus \lambda \in \mathbb{R}.^[14] The existence of an orthonormal basis of eigenvectors follows by induction on the dimension n of the underlying complex vector space \mathbb{C}^n. The base case n = 1 is immediate, as a $1 \times 1 Hermitian matrix is real and thus its single entry is both the eigenvalue and eigenvector (up to scaling). For the inductive step, assume the result holds for dimensions less than n. The characteristic polynomial \det(H - \lambda I) has degree n and thus at least one root \lambda_1 \in \mathbb{R}, yielding an eigenspace E_1. This eigenspace is orthogonal to the eigenspaces of any other distinct eigenvalues, as eigenvectors x \in E_1 and y for \lambda_2 \neq \lambda_1 satisfy \lambda_1 \langle x, y \rangle = \langle H x, y \rangle = \langle x, H y \rangle = \lambda_2 \langle x, y \rangle, forcing \langle x, y \rangle = 0. Moreover, the orthogonal complement E_1^\perp is invariant under H, since for z \in E_1^\perp, \langle H z, x \rangle = \langle z, H x \rangle = \lambda_1 \langle z, x \rangle = 0 for all x \in E_1, so H z \in E_1^\perp. The restriction of H to E_1^\perp is Hermitian on a space of dimension less than n, so by the inductive hypothesis it admits an orthonormal eigenbasis. Combining this with an orthonormal basis for E_1 yields the full basis for \mathbb{C}^n.^[14] The eigenvalues are the roots of the characteristic polynomial and thus unique up to ordering and multiplicity. Eigenspaces corresponding to distinct eigenvalues are pairwise orthogonal, as established in the proof. The spectral decomposition itself is unique up to reordering of the eigenvalues along the diagonal of D and choice of orthonormal bases within degenerate eigenspaces.^[14] For a concrete illustration, consider the $2 \times 2 Hermitian matrix

H = \begin{pmatrix} 2 & -10 \\ -10 & 2 \end{pmatrix}.

The eigenvalues are found by solving the characteristic equation \det(H - \lambda I) = 0:

\det\begin{pmatrix} 2 - \lambda & -10 \\ -10 & 2 - \lambda \end{pmatrix} = (2 - \lambda)^2 - 100 = \lambda^2 - 4\lambda - 96 = 0,

yielding roots \lambda_1 = 12 and \lambda_2 = -8. The eigenvector for \lambda_1 = 12 satisfies (H - 12 I) v = 0, giving v_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}; for \lambda_2 = -8, v_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}. Normalizing these to unit length produces

u_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \quad u_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}.

The unitary matrix is U = \begin{pmatrix} u_1 & u_2 \end{pmatrix}, and D = \diag(12, -8), so H = U D U^*. These eigenvectors are orthogonal, as \langle u_1, u_2 \rangle = 0.^[15] The spectral decomposition implies key properties of Hermitian matrices. The trace equals the sum of the eigenvalues: \tr(H) = \sum_i \lambda_i, since trace is similarity-invariant and \tr(D) = \sum \lambda_i. Similarly, the determinant is the product of the eigenvalues: \det(H) = \prod_i \lambda_i, as determinant is also similarity-invariant and \det(D) = \prod \lambda_i. These relations hold for any diagonalizable matrix but underscore the real, ordered nature of the spectrum for Hermitian matrices.^[14]

Normal matrices

A normal matrix N over the complex numbers is a square matrix that commutes with its conjugate transpose, satisfying N N^* = N^* N.^[3] The spectral theorem for normal matrices states that N is unitarily diagonalizable, meaning there exists a unitary matrix U and a diagonal matrix D such that N = U D U^*, where the diagonal entries of D are the (possibly complex) eigenvalues of N.^[16] This decomposition extends the spectral theorem for Hermitian matrices, which form a special subclass of normal matrices where the eigenvalues are real.^[17] The proof relies on the Schur decomposition, which triangularizes any matrix via a unitary similarity: N = U T U^* with T upper triangular. For normal N, the triangular form T must also be normal, and a normal upper triangular matrix is necessarily diagonal, yielding the desired D.^[16] This follows from the property that normality implies orthogonal invariant subspaces, allowing construction of an orthonormal basis of eigenvectors.^[3] Unlike Hermitian matrices, where eigenvalues are real, normal matrices can have complex eigenvalues, though their moduli |\lambda| are preserved under unitary similarity transformations, reflecting the unitary invariance of the operator norm for normal operators.^[17] To compute the decomposition, first obtain the Schur form via algorithms like the QR iteration, then verify diagonality (which holds if and only if the matrix is normal) to extract eigenvalues and eigenvectors.^[16] A representative example is the 2D rotation matrix by angle \theta,

R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},

which is normal (as it is orthogonal, hence unitary over \mathbb{C}) but not Hermitian unless \theta = 0 or \pi. Its eigenvalues are the complex pair e^{i\theta} and e^{-i\theta}, with corresponding eigenvectors involving complex entries, enabling unitary diagonalization over \mathbb{C}.^[18]

Relation to singular value decomposition

The singular value decomposition (SVD) provides a factorization for any complex m \times n matrix A, expressed as A = U \Sigma V^*, where U is an m \times m unitary matrix, V is an n \times n unitary matrix, and \Sigma is an m \times n rectangular diagonal matrix with non-negative real singular values \sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_p \geq 0 on its main diagonal (with p = \min(m, n)).^[19] This decomposition generalizes the spectral decomposition to arbitrary matrices, including non-square and non-normal ones, by revealing a "diagonal" form in an appropriate basis.^[20] For a Hermitian matrix A (where A = A^*), the spectral decomposition yields A = Q \Lambda Q^*, with Q unitary and \Lambda diagonal containing the real eigenvalues \lambda_i. In this case, the singular values of A are precisely the absolute values \sigma_i = |\lambda_i|, and the SVD takes the form A = U \Sigma V^* where \Sigma = |\Lambda| (with signs absorbed into U or V), thus recovering the spectral decomposition up to phase adjustments in the singular vectors.^[21] This connection highlights how SVD extends the unitary diagonalization of Hermitian matrices to a broader setting while preserving the non-negativity of the diagonal entries. When A is normal (satisfying A A^* = A^* A), the spectral theorem provides a unitary diagonalization A = Q \Lambda Q^* with complex eigenvalues \lambda_i on the diagonal of \Lambda. Here, the SVD aligns closely with this decomposition, as the singular values are again \sigma_i = |\lambda_i|, and the left and right singular vectors correspond to the eigenvectors of A, up to scaling by the argument of each \lambda_i.^[20] This alignment demonstrates SVD as a real-valued analog of the spectral decomposition for normal operators. Computationally, the SVD of a general matrix A is often obtained by applying the spectral decomposition to the Hermitian matrices A A^* (yielding left singular vectors and squared singular values) and A^* A (yielding right singular vectors), thereby leveraging the spectral theorem for Hermitian matrices in the process.^[22] As an illustration of how SVD generalizes diagonalization to non-square matrices, consider the $2 \times 3 matrix

X = \begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \end{pmatrix}.

Its economy SVD is X = U \Sigma V^T, where

U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \sqrt{2} & 0 \\ 0 & \sqrt{2} \end{pmatrix}, \quad V^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix},

with singular vectors derived from the eigendecompositions of X X^T and X^T X; this form diagonalizes X in the sense of revealing its action as a weighted sum of orthogonal rank-one updates, extending beyond the square-case spectral decomposition.

Infinite-dimensional case for compact operators

Self-adjoint compact operators

In the context of infinite-dimensional separable Hilbert spaces, spectral decomposition for self-adjoint compact operators provides a diagonal representation analogous to that of Hermitian matrices in finite dimensions, but with eigenvalues accumulating only at zero to account for the infinite dimensionality.^[23] A bounded linear operator K: H \to H on a Hilbert space H is called compact if the image of the closed unit ball \{x \in H : \|x\| \leq 1\} under K is relatively compact, meaning its closure is compact in the norm topology of H.^[23] Such an operator K is self-adjoint if it equals its adjoint, K = K^*, which implies that its spectrum lies on the real line and it preserves the inner product structure.^[23] The spectral theorem for these operators states that if K is a compact self-adjoint operator on a separable Hilbert space H, then there exists a countable orthonormal set \{e_n\}_{n=1}^\infty \subset H (possibly finite) consisting of eigenvectors with corresponding real eigenvalues \{\lambda_n\}_{n=1}^\infty satisfying \lambda_n \to 0 as n \to \infty, such that

Kx = \sum_{n=1}^\infty \lambda_n \langle x, e_n \rangle e_n

for every x \in H, where the series converges in the norm of H.^[23] The eigenspace for the eigenvalue 0 may have infinite multiplicity, completing the orthonormal basis of H if necessary.^[23] A proof sketch proceeds by approximating K in the operator norm by a sequence of finite-rank self-adjoint operators K_m, each of which admits a finite spectral decomposition by the finite-dimensional spectral theorem.^[24] The eigenvalues are then characterized variationally via the minimax principle: the nth largest eigenvalue satisfies

\lambda_n = \min_{\dim V = n} \max_{\substack{x \in V \\ \|x\|=1}} \langle Kx, x \rangle = \max_{\dim W = n-1} \min_{\substack{x \perp W \\ \|x\|=1}} \langle Kx, x \rangle,

allowing one to show that the eigenvalues of K_m converge to those of K and that the corresponding eigenspaces stabilize to yield the infinite orthonormal basis.^[25] The non-zero spectrum of K consists solely of discrete eigenvalues of finite multiplicity, with 0 as the only possible accumulation point; this follows from the compactness, which ensures that non-zero spectral points cannot form continuous parts or accumulate elsewhere.^[26] A concrete example arises with Hilbert-Schmidt integral operators on L^2[a,b], defined by (Kf)(x) = \int_a^b k(x,y) f(y) \, dy where the kernel k is continuous, symmetric, and positive semi-definite, ensuring K is compact and self-adjoint.^[27] By Mercer's theorem, under these conditions, k(x,y) = \sum_{n=1}^\infty \lambda_n \phi_n(x) \phi_n(y) uniformly on [a,b] \times [a,b], where \{\lambda_n\} are the non-negative eigenvalues of K with \lambda_n \to 0 and \{\phi_n\} form an orthonormal basis of eigenfunctions in L^2[a,b].^[27]

Spectral theorem for compact operators

The spectral theorem for compact self-adjoint operators asserts that if T is a compact self-adjoint operator on a separable Hilbert space \mathcal{H}, then there exists an orthonormal basis \{ e_n \} of \mathcal{H} consisting of eigenvectors of T, with corresponding eigenvalues \{ \lambda_n \} satisfying \lambda_n \to 0 as n \to \infty, and each nonzero eigenvalue having finite multiplicity.^[24] This discrete spectrum, accumulating only at zero, distinguishes compact operators from more general bounded self-adjoint operators.^[28] The decomposition is unique in the sense that the eigenvalues \lambda_n (counted with multiplicity) and the associated orthogonal projections P_n onto the finite-dimensional eigenspaces \ker(T - \lambda_n I) are uniquely determined by T.^[23] Specifically, the spectral projections satisfy P_n P_m = \delta_{nm} P_n and \sum_n P_n = I in the strong operator topology, ensuring the representation T = \sum_n \lambda_n P_n.^[29] The series \sum_n \lambda_n P_n converges to T in the strong operator topology, meaning that for every x \in \mathcal{H}, Tx = \sum_n \lambda_n P_n x with the partial sums converging in norm.^[24] This convergence holds because the eigenvalues decay to zero and the projections are mutually orthogonal with finite ranks except possibly for the kernel.^[30] This result connects to Riesz-Fredholm theory, where the essential spectrum of any compact operator on a Banach space is precisely \{0\}, reflecting the finite-dimensional nature of eigenspaces away from zero and the possible infinite-dimensional kernel.^[31] The index of T - \lambda I is zero for \lambda \neq 0, underscoring the discrete structure enforced by compactness.^[32]

General infinite-dimensional case

Bounded self-adjoint operators

For bounded self-adjoint operators on a Hilbert space, the spectral theorem provides a canonical form that generalizes the finite-dimensional diagonalization. Specifically, every bounded self-adjoint operator A on a separable Hilbert space H admits a unique spectral measure E, which is a projection-valued measure on the Borel \sigma-algebra of the real line such that A = \int_{\sigma(A)} \lambda \, dE(\lambda), where \sigma(A) denotes the spectrum of A and the integral is understood in the strong operator topology. This resolution of the identity expresses A as an "integral" over its spectrum, weighted by the spectral projections. The theorem was established by John von Neumann in his foundational work on quantum mechanics. A key consequence is that A is unitarily equivalent to multiplication by the identity function \lambda on the Hilbert space L^2(\sigma(A), \mu), where \mu is the scalar spectral measure \mu(\Delta) = \langle E(\Delta) \xi, \xi \rangle for some cyclic vector \xi \in H and Borel set \Delta \subseteq \sigma(A). This multiplication operator form highlights how the action of A corresponds to pointwise multiplication by spectral values, distributed according to the measure \mu. In this representation, the spectrum \sigma(A) is precisely the essential range of \lambda with respect to \mu. Unlike the compact case, where the spectrum is discrete and consists of eigenvalues with orthonormal eigenvectors, bounded self-adjoint operators can have continuous spectrum without point spectrum (eigenvalues). A classic example is the multiplication operator M_x on L^2[0,1] defined by (M_x f)(x) = x f(x) for f \in L^2[0,1] with Lebesgue measure. Here, \sigma(M_x) = [0,1] is purely continuous, and there are no eigenvalues: supposing M_x f = \lambda f for some \lambda \in [0,1] and f \neq 0, then f(x) = 0 almost everywhere on [0,1] \setminus \{\lambda\}, implying f = 0 almost everywhere, a contradiction. Thus, M_x has no eigenvectors, yet its spectral measure is supported on the continuous interval [0,1].^[11] The spectral measure E defines spectral subspaces via projections E(\Delta) for Borel sets \Delta \subseteq \mathbb{R}, where E(\Delta) is the orthogonal projection onto the closure of the span of generalized eigenvectors corresponding to eigenvalues in \Delta \cap \sigma(A); these subspaces satisfy E(\Delta_1) E(\Delta_2) = E(\Delta_1 \cap \Delta_2) and A E(\Delta) = E(\Delta) A. The operator norm satisfies \|A\| = \sup \{ |\lambda| : \lambda \in \sigma(A) \}, reflecting the boundedness. In the compact case, this framework specializes to a discrete sum over Dirac measures at eigenvalues. An illustrative example from quantum mechanics is the position operator Q on L^2(\mathbb{R}), which acts as multiplication by x and has spectral decomposition Q = \int_{\mathbb{R}} \lambda \, dE(\lambda), where E(\Delta) projects onto functions supported in \Delta \subseteq \mathbb{R}; this captures the continuous position spectrum over all real numbers.

Unbounded self-adjoint operators

In functional analysis, an unbounded self-adjoint operator A on a Hilbert space H is defined as a densely defined linear operator with domain D(A) such that D(A) is dense in H, A = A^* where A^* denotes the adjoint operator, and A has no finite operator norm, meaning \sup_{\|x\|=1, x \in D(A)} \|Ax\| = \infty.^[33] This contrasts with bounded self-adjoint operators, where the domain is the entire space and the norm is finite; here, the domain restriction is essential to ensure self-adjointness.^[34] The spectral theorem for unbounded self-adjoint operators extends the finite-dimensional diagonalization to infinite dimensions, stating that A is unitarily equivalent to multiplication by the independent variable \lambda on a direct sum of copies of L^2(\sigma(A), \mu), where \sigma(A) \subseteq \mathbb{R} is the spectrum of A and \mu is a \sigma-finite measure.^[35] Specifically, there exists a unitary operator U: H \to L^2(\sigma(A), \mu) such that U A U^{-1} f(\lambda) = \lambda f(\lambda) for functions f in the domain \{f \in L^2(\sigma(A), \mu) \mid \int_{\sigma(A)} |\lambda f(\lambda)|^2 \, d\mu(\lambda) < \infty\}.^[33] This representation highlights how the domain of A corresponds to functions where the multiplication remains square-integrable after weighting by \lambda, accommodating the unbounded nature of the spectrum, which typically extends to \pm \infty. Spectral multiplicity arises from the direct sum structure, where the number of copies reflects the dimension of the eigenspaces or generalized eigenspaces at each \lambda, while the essential spectrum consists of those \lambda \in \sigma(A) that are either accumulation points of the spectrum or eigenvalues of infinite multiplicity, remaining invariant under compact perturbations.^[36] For symmetric but not necessarily self-adjoint operators, the Friedrichs extension provides a canonical way to obtain a self-adjoint extension, particularly for positive symmetric operators. If T is a densely defined symmetric operator on H with \langle Tx, x \rangle \geq 0 for all x \in D(T), the Friedrichs extension is constructed by considering the quadratic form q(x) = \langle Tx, x \rangle on the form domain, completing it to a new Hilbert space, and defining the extension via the associated sesquilinear form, yielding a positive self-adjoint operator that extends T.^[37] This extension, introduced by Friedrichs, ensures the resulting operator is self-adjoint and preserves the positivity, facilitating the application of the spectral theorem.^[37] A classic example is the differential operator A = -\frac{d^2}{dx^2} on the Hilbert space L^2[0, \pi] with Dirichlet boundary conditions, defined on the domain D(A) = \{f \in H^2[0, \pi] \cap H_0^1[0, \pi] \mid f(0) = f(\pi) = 0\}. This operator is unbounded and self-adjoint, with spectral decomposition given by eigenvalues \lambda_n = n^2 and corresponding orthonormal eigenfunctions \phi_n(x) = \sqrt{\frac{2}{\pi}} \sin(nx) for n = 1, 2, \dots, so that any f \in L^2[0, \pi] expands as f(x) = \sum_{n=1}^\infty c_n \phi_n(x) with Af = \sum_{n=1}^\infty n^2 c_n \phi_n(x) for f \in D(A).^[38] Here, the spectrum is discrete \{n^2 \mid n \in \mathbb{N}\}, with essential spectrum empty and multiplicity one for each eigenvalue, illustrating how boundary conditions enforce self-adjointness and determine the spectral properties.^[38]

Spectral measures and functional calculus

In the spectral theorem for self-adjoint operators on a Hilbert space, the projection-valued measure, also known as the spectral measure E, plays a central role. It is a family of orthogonal projections \{E(\Delta)\}_{\Delta \in \mathcal{B}(\mathbb{R})}, where \mathcal{B}(\mathbb{R}) denotes the Borel \sigma-algebra on \mathbb{R}, such that E(\mathbb{R}) = I (the identity operator) and, for disjoint Borel sets \Delta_1, \Delta_2 \in \mathcal{B}(\mathbb{R}), E(\Delta_1 \cup \Delta_2) = E(\Delta_1) + E(\Delta_2). This measure is countably additive in the strong operator topology, meaning that for any countable collection of pairwise disjoint Borel sets \{\Delta_n\}_{n=1}^\infty, \sum_{n=1}^\infty E(\Delta_n) = E\left(\bigcup_{n=1}^\infty \Delta_n\right) strongly. The self-adjoint operator A is then represented as A = \int_{\mathbb{R}} \lambda \, dE(\lambda), where the integral is understood in the strong sense.^[39]^[40] The functional calculus arises naturally from the spectral measure E. For any Borel measurable function f: \mathbb{R} \to \mathbb{C}, the operator f(A) is defined by

f(A) = \int_{\mathbb{R}} f(\lambda) \, dE(\lambda),

where the integral is again strong. This construction extends the polynomial functional calculus, as polynomials in A correspond to polynomial functions of \lambda, and it preserves the algebraic structure: if f and g are Borel functions, then f(A)g(A) = (fg)(A) and f(A) + g(A) = (f + g)(A). Moreover, f(A) commutes with A for any such f, and if f is real-valued, f(A) is self-adjoint. For unbounded self-adjoint operators, the domain of f(A) consists of those vectors \xi \in H for which \int_{\mathbb{R}} |f(\lambda)|^2 \, d\|E(\lambda)\xi\|^2 < \infty.^[39]^[41] Stone's formula provides an explicit way to recover the spectral projections from the resolvent for intervals. For -\infty < a < b < \infty,

E((a,b)) = \frac{1}{2\pi i} \lim_{\varepsilon \to 0^+} \int_a^b \left[ R(\lambda + i\varepsilon, A) - R(\lambda - i\varepsilon, A) \right] d\lambda,

with adjustments for the endpoints to obtain projections like E((a,b]).^[40] For an isolated eigenvalue \lambda \in \mathbb{R}, the projection onto the corresponding eigenspace is given by

E(\{\lambda\}) = \frac{1}{2\pi i} \int_{\gamma} R(z, A) \, dz,

where R(z, A) = (A - zI)^{-1} is the resolvent operator and \gamma is a closed contour enclosing \lambda but no other points of the spectrum.^[42] The direct integral representation offers another perspective on the spectral decomposition. The Hilbert space H is unitarily equivalent to a direct integral \int^\oplus_{\sigma(A)} H_\lambda \, d\mu(\lambda), where \sigma(A) is the spectrum of A, \mu is a suitable measure on \sigma(A), and each H_\lambda is a Hilbert space fiber. In this representation, A acts as multiplication by the independent variable \lambda on the direct integral, and the spectral measure E corresponds to integration over these fibers.^[43] Regarding multiplicity, a self-adjoint operator A is called simple if there exists a cyclic vector \xi \in H such that the closed linear span of \{p(A)\xi \mid p \text{ [polynomial](/page/Polynomial)}\} is all of H, or equivalently, the scalar spectral measure \langle E(\cdot)\xi, \xi \rangle has full support on \sigma(A). In general, the multiplicity function m: \mathbb{R} \to \mathbb{N} \cup \{\infty\} describes the dimension of the fibers H_\lambda in the direct integral, capturing the "degeneracy" of the spectrum almost everywhere with respect to the spectral measure.^[44]

Applications

In quantum mechanics

In quantum mechanics, physical observables such as position, momentum, and energy are represented by self-adjoint operators on a Hilbert space, ensuring that their eigenvalues are real numbers corresponding to the possible outcomes of measurements.^[45] The spectrum of such an operator, consisting of its eigenvalues, determines the set of measurable values for the observable.^[46] The spectral decomposition theorem provides a rigorous framework for these operators, expressing a self-adjoint operator A as A = \int \lambda \, dE(\lambda), where E is a projection-valued measure (spectral measure) over the Borel sets of the real line, with the integral taken with respect to the spectral resolution.^[47] For a quantum state represented by a unit vector \psi in the Hilbert space, the probability of obtaining a measurement outcome in a Borel set \Delta \subseteq \mathbb{R} is given by \|E(\Delta) \psi\|^2 = \langle \psi | E(\Delta) | \psi \rangle, which quantifies the likelihood associated with the projection onto the corresponding eigensubspace.^[48] In the Heisenberg picture, where states are time-independent and operators evolve, the time evolution of an observable A is governed by the unitary group generated by the Hamiltonian H, yielding A(t) = e^{i t H / \hbar} A e^{-i t H / \hbar}. This evolution leverages the functional calculus from the spectral theorem, allowing functions of self-adjoint operators like f(A) = \int f(\lambda) \, dE(\lambda) to define dynamics, such as the time-dependent propagator for the observable itself.^[47] A canonical example is the quantum harmonic oscillator, whose Hamiltonian is H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2, where p and x are the momentum and position operators (unbounded self-adjoint operators). Using ladder operators a = \sqrt{\frac{m \omega}{2 \hbar}} \left( x + \frac{i p}{m \omega} \right) and a^\dagger = \sqrt{\frac{m \omega}{2 \hbar}} \left( x - \frac{i p}{m \omega} \right), the Hamiltonian decomposes as H = \hbar \omega \left( a^\dagger a + \frac{1}{2} \right), revealing a discrete spectrum of eigenvalues (n + \frac{1}{2}) \hbar \omega for n = 0, 1, 2, \dots, with corresponding energy eigenstates forming a complete orthonormal basis.^[49] The projections E(\Delta) in the spectral measure directly connect to the collapse postulate of measurement, where upon observing an outcome in \Delta, the state collapses to the normalized projection E(\Delta) \psi / \|E(\Delta) \psi\|, reflecting the irreversible update of the quantum state consistent with the probabilistic interpretation.^[48]

In numerical analysis and signal processing

In numerical analysis, spectral decomposition of Hermitian matrices is computed using stable iterative algorithms that exploit the matrix's symmetry to ensure real eigenvalues and orthogonal eigenvectors. The QR algorithm, adapted for Hermitian matrices, iteratively applies QR factorizations with shifts to converge to the diagonal form, achieving quadratic convergence near eigenvalues. This method forms the basis for implementations in libraries like LAPACK, where it computes the full eigendecomposition for dense matrices up to moderate sizes (e.g., n ≈ 1000). For large sparse symmetric matrices, the Lanczos algorithm generates a tridiagonal matrix via orthogonal projections, reducing the problem to a smaller dense eigendecomposition while approximating extremal eigenvalues efficiently. It requires O(k n) operations for k iterations on an n × n matrix, making it suitable for applications like structural engineering simulations where only a few eigenvalues are needed. Reorthogonalization techniques mitigate loss of orthogonality in the Lanczos vectors, ensuring numerical stability. Error analysis in spectral decomposition emphasizes the conditioning of eigenvalues, which measures sensitivity to perturbations: for a simple eigenvalue λ_i, the relative perturbation bound is approximately κ(V) / |λ_i - λ_j| for the closest distinct eigenvalue λ_j, where κ(V) is the condition number of the eigenvector matrix. The Bauer-Fike theorem provides a disk-based bound, stating that each perturbed eigenvalue lies within a disk of radius n cond(Q) ε centered at an original eigenvalue, where Q diagonalizes the matrix and ε is the perturbation norm. This quantifies risks in finite-precision arithmetic, guiding the use of Wilkinson's shift strategy to improve stability. In signal processing, the Fourier transform represents the spectral decomposition of the translation operator on L^2 functions, diagonalizing it with eigenfunctions as complex exponentials and eigenvalues as frequencies, enabling efficient frequency-domain analysis via the fast Fourier transform (FFT). Principal component analysis (PCA) applies eigendecomposition to the covariance matrix of data, projecting onto directions of maximum variance; the eigenvalues quantify explained variance, with the first few often capturing over 90% in high-dimensional datasets like images. Briefly, this relates to singular value decomposition for centered data. A practical example is JPEG image compression, where the discrete cosine transform (DCT) performs block-wise spectral decomposition, approximating the Karhunen-Loève transform for stationary processes and related to the eigendecomposition of circulant matrices via cosine basis functions. Low-frequency coefficients are retained after quantization, achieving compression ratios up to 20:1 with minimal perceptual loss, as higher frequencies correspond to fine details. Challenges arise with ill-conditioned matrices, where clustered eigenvalues amplify rounding errors, potentially causing O(ε n^{3/2}) relative perturbations in computed eigenvalues, as analyzed in backward error bounds. Deflation techniques address this by isolating converged eigenpairs—subtracting rank-one updates or using implicit shifts—allowing accurate computation of remaining spectrum without full recomputation, essential for multiple eigenvalues in applications like vibration analysis.

Extensions to other decompositions

Spectral decomposition generalizes to non-normal operators via the Jordan canonical form, which replaces diagonalization with a block structure consisting of Jordan blocks corresponding to defective eigenvalues, where each block features the eigenvalue on the diagonal and ones on the superdiagonal.^[50] This form expresses the operator as A = P J P^{-1}, with J upper triangular and P the matrix of generalized eigenvectors, extending the diagonal case of the spectral theorem for normal operators.^[50] Another extension is the polar decomposition, which factors any square matrix A as A = U |A|, where U is unitary and |A| = \sqrt{A^* A} is positive semi-definite; this links directly to the singular value decomposition (SVD), as the SVD provides the explicit form of |A| via its singular values and U via the corresponding unitary factors.^[51] For operators with continuous spectrum, wavelet and frame decompositions offer generalized spectral bases, using overcomplete, non-orthogonal systems to represent signals or functions in a manner analogous to eigenbasis expansions but accommodating redundancy and localization in both time and frequency domains.^[52] These approaches, rooted in frame theory, enable decompositions for multiplication operators or integral operators where point eigenvalues are absent.^[52] The foundational work on the infinite-dimensional spectral theorem for normal operators in Hilbert space was developed by John von Neumann in 1929, unifying earlier finite-dimensional results with operator theory in quantum mechanics.^[53] Despite these extensions, limitations persist: not all bounded operators admit a diagonalizing spectral decomposition, particularly non-normal ones, where the Jordan form reveals algebraic multiplicity exceeding geometric multiplicity.^[54] For such operators, pseudospectra—regions where the spectrum is sensitive to perturbations—extend the analysis beyond eigenvalues, quantifying transient growth and instability not captured by the point spectrum alone.^[54]

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