5.2 Spectral theorem for self-adjoint and normal operators

The spectral theorem for self-adjoint and normal operators is a game-changer in linear algebra. It shows that these operators have a complete set of orthonormal eigenvectors, letting us break them down into simpler parts.

This theorem is key to understanding how operators work on inner product spaces. It connects abstract math to real-world applications, especially in quantum mechanics where it helps explain how we measure physical properties of particles.

Spectral theorem for operators

Fundamental concepts of the spectral theorem

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• Spectral theorem asserts every self-adjoint or normal operator on a finite-dimensional inner product space has an orthonormal basis of eigenvectors
• Self-adjoint operator T yields real eigenvalues with orthogonal eigenvectors for distinct eigenvalues
• Normal operator N produces orthogonal eigenvectors for distinct eigenvalues, allowing complex eigenvalues
• Guarantees existence of unitary matrix U diagonalizing the operator (UTU or UNU)
• Diagonal entries of resulting matrix represent eigenvalues of original operator
• Columns of unitary matrix U comprise corresponding orthonormal eigenvectors
• Generalizes diagonalization process for symmetric matrices to self-adjoint and normal operators

Mathematical formulation and properties

• Spectral decomposition for self-adjoint operator T: $T = \sum_{i=1}^n \lambda_i P_i$
  • $\lambda_i$ denote eigenvalues
  • $P_i$ represent orthogonal projections onto eigenspaces
• For normal operator N: $N = \sum_{i=1}^n \lambda_i P_i$
  • Eigenvalues $\lambda_i$ may be complex
• Orthogonality of eigenvectors: $\langle v_i, v_j \rangle = 0$ for $i \neq j$
• Completeness of eigenbasis: $\sum_{i=1}^n P_i = I$ (identity operator)
• Unitary diagonalization: $T = UDU^*$ or $N = UDU^*$
  • U unitary matrix (columns orthonormal eigenvectors)
  • D diagonal matrix (entries eigenvalues)

Proof of the spectral theorem

Proof strategy for self-adjoint operators

• Begin by proving existence of at least one eigenvector using extremal principle on quadratic form $\langle Tx,x \rangle$
• Demonstrate orthogonal complement of eigenspace invariant under operator T
• Employ induction on vector space dimension to extend result to full set of eigenvectors
• Construct orthonormal basis of eigenvectors using Gram-Schmidt process if necessary

Proof elements for normal operators

• Prove $(N^*N - NN^*)v = 0$ for any eigenvector v of N
• Show eigenvectors corresponding to distinct eigenvalues orthogonal for both self-adjoint and normal operators
• Utilize properties of normal operators: $\|Nx\| = \|N^*x\|$ for all vectors x
• Demonstrate commutativity of N and N* implies simultaneous diagonalizability

Key steps in the proof

• Establish existence of eigenvalues using characteristic polynomial: $det(T - \lambda I) = 0$
• Prove reality of eigenvalues for self-adjoint operators: $\lambda = \frac{\langle Tv,v \rangle}{\langle v,v \rangle}$ for eigenvector v
• Show orthogonality of eigenvectors: $\langle Tv_1,v_2 \rangle = \lambda_1 \langle v_1,v_2 \rangle = \langle v_1,Tv_2 \rangle = \lambda_2 \langle v_1,v_2 \rangle$
• Construct unitary matrix U using normalized eigenvectors as columns
• Verify diagonalization: $U^*TU = D$ or $U^*NU = D$
• Conclude by expressing original operator as $T = UDU^*$ or $N = UDU^*$

Diagonalization using the spectral theorem

Process for diagonalizing self-adjoint operators

• Identify operator as self-adjoint by verifying $T^* = T$
• Calculate characteristic polynomial $det(T - \lambda I) = 0$ to find eigenvalues
• For each eigenvalue, determine corresponding eigenvectors: $(T - \lambda I)v = 0$
• Normalize eigenvectors to unit length: $u = \frac{v}{\|v\|}$
• Ensure orthonormal set for repeated eigenvalues (use Gram-Schmidt if needed)
• Construct unitary matrix U using normalized eigenvectors as columns
• Compute diagonal matrix $D = U^*TU$ to verify diagonalization
• Express original operator as $T = UDU^*$ completing diagonalization process

Diagonalization of normal operators

• Confirm operator normality by checking $N^*N = NN^*$
• Find eigenvalues through characteristic equation $det(N - \lambda I) = 0$
• Determine eigenvectors for each eigenvalue solving $(N - \lambda I)v = 0$
• Normalize and orthogonalize eigenvectors (Gram-Schmidt process for degenerate eigenvalues)
• Form unitary matrix U with orthonormal eigenvectors as columns
• Verify diagonalization by computing $D = U^*NU$
• Represent normal operator as $N = UDU^*$

Examples and applications

• Diagonalize Hermitian matrix: $A = \begin{pmatrix} 2 & i \\ -i & 2 \end{pmatrix}$
• Spectral decomposition of rotation operator in 2D: $R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
• Apply spectral theorem to solve differential equations: $\frac{d^2y}{dx^2} + y = 0$ using eigenfunctions of d²/dx²
• Analyze vibration modes of a system using spectral decomposition of mass and stiffness matrices

Implications of the spectral theorem in quantum mechanics

Observables and measurements

• Quantum observables represented by self-adjoint operators ensuring real eigenvalues (physical measurements)
• Eigenvalues correspond to possible measurement outcomes (energy levels, spin states)
• Eigenvectors represent pure states yielding definite measurement values
• State space decomposition into orthogonal subspaces for distinct measurement outcomes
• Probabilistic nature reflected in state vector expansion using observable's eigenvectors
• Measurement postulate explained mathematically through spectral theorem (wave function collapse)

Applications in quantum systems

• Time evolution analysis: $\psi(t) = e^{-iHt/\hbar}\psi(0)$ using spectral decomposition of Hamiltonian H
• Computational methods in quantum chemistry (molecular orbital theory)
• Solid-state physics applications (band structure calculations)
• Quantum information theory: qubit representations and operations
• Perturbation theory development using spectral theorem as foundation

Examples in physical systems

• Hydrogen atom energy levels derived from spectral analysis of Hamiltonian
• Stern-Gerlach experiment explained through spin operator eigenstates
• Harmonic oscillator energy states as eigenfunctions of Hamiltonian
• Angular momentum quantization from eigenvalues of L² and Lz operators
• Zeeman effect analysis using perturbation of Hamiltonian eigenvalues