Matrices, eigenvalues, and inner product spaces are the language of quantum mechanics, classical mechanics (normal modes), and every numerical simulation. This chapter develops the tools physicists use most.
Identify vector spaces and inner product spaces, and apply the Cauchy-Schwarz inequality.
Find eigenvalues and eigenvectors of a matrix by solving the characteristic equation.
State the spectral theorem for Hermitian operators and explain its role in quantum mechanics.
Compute the singular value decomposition and use it for low-rank approximation and least-squares fitting.
Determine whether two Hermitian operators share an eigenbasis by checking if they commute.
LA.1 Vector Spaces and Inner Products
Definition LA.1 — Vector Space
A vector space V over a field F (usually ℝ or ℂ) is a set with addition and scalar multiplication satisfying 8 axioms (commutativity, associativity, distributivity, identity, inverses). Key examples in physics:ℝⁿ:column vectors, forces, positions. ℂⁿ: quantum states (spinors, finite-dimensional Hilbert spaces).L2(a,b: square-integrable functions — the Hilbert space of quantum wavefunctions.Polynomials, matrices, operators — all are vector spaces under appropriate operations.
An inner product ⟨u, v⟩ on a complex vector space satisfies: linearity in v, conjugate-linearity in u, ⟨v,v⟩ ≥ 0 with equality iff v = 0. For ℂⁿ: ⟨u, v⟩ = u†v = Σᵢ uᵢ* vᵢ. For L²: ⟨f, g⟩ = ∫ f*(x) g(x) dx.
⟨u,v⟩≤∥u∥∥v∥(Cauchy−Schwarzinequality)(LA.1)
LA.2 Eigenvalues and Eigenvectors
For a linear operator A: V → V, a nonzero vector v with Av = λv is an eigenvector with eigenvalue λ. Eigenvalues satisfy:
det(A−\lambdaI)=0(characteristicequation)(LA.2)
For an n×n matrix, this is an nth-degree polynomial — giving n eigenvalues (counting multiplicity). The eigenvalues determine: stability of linear systems (Re(λ) < 0 → stable), principal stresses in mechanics, normal mode frequencies, and the spectrum of quantum observables.
Example LA.1 — Normal Modes of Two Coupled Oscillators
An operator A is Hermitian (self−adjoint)ifA=A†(Aequalsitsconjugatetranspose).Keyproperties1.Alleigenvaluesarereal(provedby⟨v∣A∣v⟩=⟨Av∣v⟩→λ=λ∗2. Eigenvectors for different eigenvalues are orthogonal3.(Spectraltheorem)Ahasacompleteorthonormaleigenbasis:A=∑iλi∣vi⟩⟨vi∣In quantum mechanics, observables are Hermitian operators — this guarantees real measured values and allows complete state expansion in the eigenstate basis.
Analogously, unitary operators U satisfy U†U = I (preserve inner products). All eigenvalues lie on the unit circle (|λ| = 1). Time evolution e^(−iHt/ℏ) is unitary when H is Hermitian — unitarity is quantum mechanical probability conservation.
LA.4 Singular Value Decomposition
Every m×n matrix A can be written as:
A=U\sumV†(SVD)(LA.3)
where U (m×m) and V (n×n) are unitary and Σ is diagonal with non-negative real entries σ₁ ≥ σ₂ ≥ ... ≥ 0 (singular values). The number of nonzero σᵢ is the rank of A. SVD is the foundation of:
Principal Component Analysis (PCA): the columns of V are principal axes — directions of maximum variance in data.
Low-rank approximation: A ≈ U_kΣ_kV_k† (keep only k largest singular values) — used in image compression, recommendation systems, quantum state compression.
Least squares: the pseudoinverse A⁺ = VΣ⁺U† solves the least-squares problem min‖Ax − b‖ when A is non-square.
Theorem LA.1 — Spectral Theorem for Commuting Observables
Two Hermitian operators A and B can be simultaneously diagonalized(shareacompleteorthonormaleigenbasis)ifandonlyiftheycommute:[A,B]=les can be simultaneously measured(sharpvalues)onlyiftheycommute.Positionandmomentumdon′tcommute([x^,p^]=arp.L2andLzcommute—bothcanbesimultaneouslyspecified.
Basis vectors are choices: physical vectors and states are independent of representation.
Eigenvectors can be degenerate: a degenerate eigenspace needs a basis choice.
Hermitian operators have real eigenvalues: this is why they represent observables in quantum mechanics.
Matrix multiplication is not commutative: operator order can change the result.
Exercises — LA.1–LA.4 Linear Algebra for Physics
1.
FindthelargereigenvalueofthematrixA=[[4,1],[2,3.
Straightforward
2.
FindtherankofthematrixA=[[1,2,3],[2,4,6.
Straightforward
3.ApplyGram−Schmidtorthogonalizationtothevectorsv1=(1,1,0),v2=(1,0,1),v3=(0,1,1)to produce an orthonormal basis.
Straightforward
4.Write the matrix representations of the harmonic oscillator Hamiltonian H and position operator x in the energy eigenstate basis {|0⟩, |1⟩, |2⟩, |3⟩}. What selection rule does this imply?
Intermediate
5.UsetheCayley−HamiltontheoremtofindarecursionforAnwhereA=[[1,1],[1,0]].Howdoes this relate to Fibonacci numbers?
Intermediate
6.DiscussthepositionandmomentumoperatorsasoperatorsonL2(R).Whydotheynothavenormalizable eigenstates? How does the continuous spectrum relate to the uncertainty principle?