Matrices, eigenvalues, and diagonalisation

The operator is the physical object; the matrix is its appearance in a chosen basis. Insert two resolutions of the identity, A = Σᵢⱼ |i⟩⟨i|A|j⟩⟨j|, and the n² numbers Aᵢⱼ = ⟨i|A|j⟩ are everything the basis (|i⟩) can know about A. Move to another orthonormal basis with a unitary U and the array transforms by similarity, A′ = U†AU — different entries, identical trace, determinant and eigenvalues, because those belong to the operator, not to the representation. Concretely: S_z is (ħ/2)⋅diag(1, −1) in its own eigenbasis, but written in the eigenbasis of Sₓ the same operator becomes (ħ/2)[[0, 1],[1, 0]] — nothing physical changed, only the coordinates. Off-diagonal entries are the basis-dependent statement that A maps |j⟩ partly onto |i⟩; a matrix is diagonal precisely in a basis built from its own eigenvectors.

An eigenvector is a direction the operator does not rotate, only rescales, and the eigenvalue a is the factor. In quantum mechanics the equation carries the measurement content of the theory: measure A on its eigenstate |ψ⟩ and the result is a with certainty; measure it on anything else and the possible results are still only the eigenvalues. A nonzero solution of (A − aI)|ψ⟩ = 0 exists exactly when A − aI is singular, so the eigenvalues are the roots of det(A − aI) = 0 — degree n, hence n roots over ℂ counted with multiplicity. Two invariants come free and check every hand calculation: the eigenvalues sum to the trace and multiply to the determinant. For any traceless 2×2 with determinant −1 — all three Pauli matrices — that alone forces a₊ + a₋ = 0 and a₊a₋ = −1, so a = ±1 with no polynomial solved at all.

Write the general 2×2 Hermitian matrix as H = [[α, β],[β*, γ]] with α, γ real, and name the half-trace m = (α+γ)/2 and half-gap d = (α−γ)/2. The characteristic polynomial collapses to (a − m)² = d² + |β|², so a± = m ± √(d² + |β|²) — memorise it, because spin-½ in a magnetic field, the two-level atom and every qubit Hamiltonian to come are this line. For real β the eigenvectors are (cos θ, sin θ) and (−sin θ, cos θ) with tan 2θ = 2β/(α−γ). Numbers: H = [[5.0, 1.5],[1.5, 1.0]] has m = 3.0 and d = 2.0, so a± = 3.0 ± √(4.0 + 2.25) = 5.5 and 0.5. The formula also carries a structural lesson: the splitting 2√(d² + |β|²) is at least 2|β|, so as any parameter sweeps the diagonal through resonance the two levels close to 2|β| and repel rather than cross — the avoided crossing drawn in the figure below.

The spectral theorem is the payoff for demanding A† = A: every eigenvalue is real, eigenvectors of distinct eigenvalues are automatically orthogonal, and even a degenerate eigenvalue supplies a full eigenspace inside which Gram-Schmidt builds an orthonormal set — so a complete orthonormal eigenbasis always exists and U†AU = diag(a₁,…, aₙ). Drop the hypothesis and diagonalisation can fail outright. Take [[1, 1],[0, 1]]: the characteristic polynomial (1 − a)² promises the eigenvalue 1 twice, but A − I = [[0, 1],[0, 0]] has rank 1, so its null space is the single line through (1, 0). Algebraic multiplicity 2 against geometric multiplicity 1 is the definition of a defective matrix, and no similarity transform will ever make it diagonal. The exact boundary of unitary diagonalisability is normality, [A, A†] = 0, which also admits unitary and anti-Hermitian matrices; observables sit safely inside it by postulate.

By hand the characteristic polynomial ends at 3×3; numerically it should never be formed at all, because polynomial roots respond violently to tiny changes in coefficients — Wilkinson's example perturbs one coefficient of a degree-20 polynomial by 2⁻²³ and scatters its roots across the complex plane. Library routines instead apply a sequence of unitary similarity transforms to the matrix itself, which cannot amplify error because unitaries preserve norms. numpy.linalg.eigh takes a Hermitian matrix and returns ascending real eigenvalues with orthonormal eigenvector columns; numpy.linalg.eig handles the general case, where eigenvalues may be complex, columns need not be orthogonal, and a nearly defective matrix betrays itself through nearly parallel eigenvectors. One trap deserves a label: eigh reads only one triangle of the array and assumes symmetry for the rest, so a matrix that is not actually Hermitian is silently replaced by one that is. Check max|A − A†| before calling, and ‖Av − av‖ after.

Diagonalisation is a one-time cost that makes everything after it componentwise. In its eigenbasis A = Σ aₖ|k⟩⟨k|, so powers, inverses (when no aₖ vanishes) and arbitrary functions come by acting on the spectrum alone: f(A) = Σ f(aₖ)|k⟩⟨k| — the identity that will define exp(−iHt/ħ) in Unit 4 without summing a single matrix series. A state expands as |ψ⟩ = Σ cₖ|k⟩ with cₖ = ⟨k|ψ⟩, and |cₖ|² is the probability that a measurement returns aₖ. One honesty clause: a degenerate eigenvalue of multiplicity m fixes an m-dimensional eigenspace, not particular vectors, so any orthonormal frame inside it is equally an answer. When eigh returns eigenvalues [0, 3, 3] for the matrix in the hard example below, its two a = 3 columns are one legitimate choice among a continuum — compare the subspaces they span, never the columns entry by entry.