Spectral Theorem

Introduction

The spectral theorem is a cornerstone result in linear algebra relating normal linear operators on finite-dimensional inner product spaces to their eigenvalues and eigenvectors. It guarantees diagonalization via an orthonormal basis of eigenvectors, facilitating matrix simplification and functional calculus. Fundamental to quantum mechanics, numerical analysis, and differential equations, it bridges abstract theory with computational methods.

"The spectral theorem provides the key to unlocking the structure of linear transformations through their spectra." -- Gilbert Strang

Preliminaries

Vector Spaces and Inner Products

Definition: vector space over ℝ or ℂ with inner product ⟨·,·⟩. Properties: linearity, positivity, conjugate symmetry. Induces norm and orthogonality.

Linear Operators and Matrices

Linear operator T: V→V. Matrix representation depends on basis. Changing basis alters matrix but preserves operator properties.

Eigenvalues and Eigenvectors

Eigenvalue λ: scalar satisfying T(v) = λv, v≠0 eigenvector. Spectrum σ(T) = set of eigenvalues. Algebraic and geometric multiplicities.

Adjoint Operators

For T on inner product space, adjoint T* satisfies ⟨T(v), w⟩ = ⟨v, T*(w)⟩ ∀ v,w. Matrix adjoint = conjugate transpose.

Normal Operators

Definition

Operator T is normal if T*T = TT*. Includes self-adjoint, unitary, and skew-adjoint operators.

Properties

Normal operators are diagonalizable by unitary matrices. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

Examples

Self-adjoint: T = T*. Unitary: T*T = TT* = I. Diagonal matrices with complex entries are normal.

Statement of the Spectral Theorem

Finite-Dimensional Case

Every normal operator T on a finite-dimensional complex inner product space admits an orthonormal basis consisting of eigenvectors of T. Equivalently, T is unitarily diagonalizable.

Real Version

For real inner product spaces, every self-adjoint operator admits an orthonormal eigenbasis; normal operators may require complexification.

Matrix Formulation

For normal matrix A ∈ M_n(ℂ), there exists unitary U with U*AU = D diagonal.

A ∈ M_n(ℂ) normal ⇒ ∃ U unitary: U* A U = diag(λ₁, ..., λ_n)

Spectral Decomposition

Projection Operators

Decomposition of identity I = ∑ P_i, where each P_i is orthogonal projection onto eigenspace of λ_i.

Operator Expression

T = ∑ λ_i P_i. Functional calculus applies: f(T) = ∑ f(λ_i) P_i.

Matrix Form

A = ∑ λ_i u_i u_i*, where u_i is eigenvector with unit norm.

T = ∑_{i=1}^k λ_i P_i, P_i P_j = δ_ij P_i, ∑_{i=1}^k P_i = I

Real Symmetric Matrices

Definition

A ∈ M_n(ℝ) symmetric if A = Aᵀ. Special subclass of normal matrices.

Diagonalization

Exists orthogonal Q with Qᵀ A Q = D diagonal real.

Properties

Eigenvalues real. Eigenvectors form orthonormal basis. Spectral theorem guarantees orthogonal diagonalization.

Hermitian Matrices

Definition

A ∈ M_n(ℂ) Hermitian if A = A*. Generalizes real symmetric matrices.

Eigenvalues and Eigenvectors

Eigenvalues real. Eigenvectors orthonormal with respect to complex inner product.

Diagonalization

Exists unitary U with U* A U = D real diagonal.

Unitary Diagonalization

Definition

A matrix A is unitarily diagonalizable if ∃ unitary U such that U*AU = D diagonal.

Relation to Normality

Normality ⇔ Unitary diagonalizability over ℂ.

Consequences

Preserves inner product. Simplifies functional calculus and power computations.

A normal ⇔ ∃ unitary U: U* A U = diag(λ₁,...,λ_n)

Applications

Quantum Mechanics

Observables represented by Hermitian operators. Eigenvalues correspond to measurable quantities.

Numerical Linear Algebra

Matrix diagonalization for efficient computation of powers, exponentials, and functions.

Differential Equations

Diagonalization simplifies systems of ODEs and PDEs with symmetric operators.

Data Science

PCA uses eigen-decomposition of covariance matrices (real symmetric) for dimensionality reduction.

Examples

Real Symmetric Matrix

Matrix A = [[2,1],[1,2]] symmetric, eigenvalues 3 and 1, eigenvectors orthogonal.

Hermitian Matrix

A = [[1, i], [-i, 1]] Hermitian, eigenvalues 2 and 0, diagonalizable by unitary matrix.

Normal but Not Hermitian

Unitary matrix with complex eigenvalues on unit circle, diagonalizable but not Hermitian.

Limitations and Extensions

Non-Normal Operators

Not all operators diagonalizable; defective matrices lack full eigenbasis.

Infinite-Dimensional Spaces

Spectral theorem extends to bounded normal operators on Hilbert spaces with spectral measures.

Spectral Theorem for Unbounded Operators

Requires functional analysis framework; essential in quantum theory.

Jordan Normal Form

Generalizes diagonalization; applies to non-normal operators but loses orthogonality.

Proof Sketch

Key Idea

Use Schur decomposition to express normal operator as unitary times upper-triangular times unitary*. Show off-diagonal entries vanish.

Outline

1. Apply Schur decomposition: T = U R U* with R upper-triangular.
2. Normality: T T* = T* T implies R R* = R* R.
3. Upper-triangular normal matrices are diagonal.
4. Therefore, T diagonalizable by unitary U.

Consequences

Eigenvectors form orthonormal basis. Spectrum consists of eigenvalues on diagonal.

Given normal T:1. T = U R U*, R upper-triangular2. T normal ⇒ R normal3. Upper-triangular + normal ⇒ R diagonal4. Hence, T unitarily diagonalizable

References

• Horn, R.A., Johnson, C.R., "Matrix Analysis," Cambridge University Press, vol. 2, 2013, pp. 341-377.
• Axler, S., "Linear Algebra Done Right," Springer, 3rd ed., 2015, pp. 205-230.
• Reed, M., Simon, B., "Methods of Modern Mathematical Physics I: Functional Analysis," Academic Press, 1980, pp. 222-240.
• Strang, G., "Introduction to Linear Algebra," Wellesley-Cambridge Press, 5th ed., 2016, pp. 385-410.
• Conway, J.B., "A Course in Functional Analysis," Springer, 2nd ed., 1990, pp. 215-245.