Orthogonal Projections and Spectral Resolution

Derivation

Orthogonal Projections and Spectral Resolution

D-089 Home PU-104 Threads symmetry · chance Depends on Spectral Theorem for Self-Adjoint Operators, Properties of Hermitian Operators

Statement

On a Hilbert space H, a bounded operator P is an orthogonal projector if and only if it is both idempotent, P² = P, and self-adjoint, P† = P; such a P projects every vector onto a closed subspace M = ran P along its orthogonal complement M⊥ = ker P. Consequently, a self-adjoint operator A with eigenvalues λ_i admits the spectral resolution A = Σ_i λ_i P_i, in which the eigenprojectors satisfy P_iP_j = δ_ijP_i and resolve the identity, Σ_i P_i = I.

Why it matters

The resolution of the identity is the algebraic backbone of the measurement postulate. Every observable is decomposed into a weighted sum of orthogonal projectors, one per possible outcome; the projectors are the mathematical objects that the Born rule feeds on, turning an abstract state vector into a probability distribution over eigenvalues. This is the point where the symmetry thread (self-adjointness, a reality condition) and the chance thread (probabilities of outcomes) meet in a single equation.

Beyond quantum theory, spectral resolution underwrites the functional calculus — the ability to define f(A) for any function f by acting on eigenvalues alone — which is how time-evolution operators e^−iHt/ħ, density matrices, and thermal averages are actually computed.

Assumptions

The space is a complete inner-product space (Hilbert space).Without completeness, ran P need not be closed and the orthogonal decomposition H = M ⊕ M⊥ can fail; a "projector" onto a non-closed subspace is unbounded and the geometry breaks. P is bounded.Idempotent unbounded operators exist but are not everywhere-defined; boundedness is what lets P† be defined on all of H and equated with P.

The self-adjoint operator A has purely discrete spectrum with a complete set of eigenvectors.For operators with continuous spectrum the sum Σ_i λ_i P_i must be replaced by a Stieltjes integral ∫ λ dE(λ) over a projection-valued measure; the finite/countable resolution shown here no longer holds verbatim.

Derivation

P² = P ⇒ for any eigenvalue λ² = λ ⇒ λ ∈ {0, 1}

Applying an idempotent to its own eigenvector, Pv = λv gives P²v = λ²v = λv, so the only admissible eigenvalues are 0 and 1. A

v = Pv + (I − P)v for every v ∈ H

Trivial addition and subtraction of Pv; the first term lies in ran P, and since P(I − P)v = (P − P²)v = 0 the second lies in ker P. This gives the algebraic direct sum H = ran P ⊕ ker P before any metric is used. A

⟨Pa, (I − P)b⟩ = ⟨a, P†(I − P)b⟩ = ⟨a, P(I − P)b⟩ = ⟨a, (P − P²)b⟩ = 0

Here the self-adjoint hypothesis P† = P enters: move P across the inner product, then use idempotency. Every vector in ran P is orthogonal to every vector in ker P, upgrading the direct sum of Step 2 to an orthogonal decomposition. B

⟨v, Pv⟩ = ⟨v, P²v⟩ = ⟨P†v, Pv⟩ = ⟨Pv, Pv⟩ = ‖Pv‖² ≥ 0

Combining P² = P and P† = P shows P is a positive operator and that ‖Pv‖² = ⟨v, Pv⟩ ≤ ‖v‖ ‖Pv‖ by Cauchy–Schwarz, hence ‖Pv‖ ≤ ‖v‖: an orthogonal projector is a norm-one contraction, geometrically the "nearest point in M" map. C

A self-adjoint ⇒ λ_i ∈ ℝ, and λ_i ≠ λ_j ⇒ ran P_i ⊥ ran P_j

From hermitian-operator-properties: eigenvalues of a self-adjoint operator are real and eigenvectors belonging to distinct eigenvalues are orthogonal. Define P_i as the orthogonal projector onto the eigenspace of λ_i; distinctness gives P_iP_j = δ_ijP_i. B

Σ_i P_i = I (resolution of the identity)

The spectral theorem (spectral-theorem-self-adjoint) guarantees the eigenvectors form a complete orthonormal basis of H; the projectors onto the mutually orthogonal eigenspaces therefore sum to the identity, since every v is reconstructed as v = Σ_i P_iv. C

A = A I = A Σ_i P_i = Σ_i A P_i = Σ_i λ_i P_i

Insert the resolution of the identity to the right of A, then use that P_i projects onto the λ_i-eigenspace where A acts as multiplication by λ_i, so A P_i = λ_iP_i. This is the spectral resolution. B

f(A) = Σ_i f(λ_i) P_i (functional calculus)

Because P_iP_j = δ_ijP_i, powers collapse as Aⁿ = Σ_i λ_iⁿ P_i; extending linearly over polynomials and closing under uniform limits defines f(A) for continuous f by acting on eigenvalues alone. C

Result

P² = P , P† = P ⟹ H = ran P ⊥⊕ ker P A = Σ_i λ_i P_i , Σ_i P_i = I , P_iP_j = δ_ijP_i

Reading. A self-adjoint idempotent is exactly an orthogonal projection: it splits the space into a subspace it keeps untouched and the perpendicular subspace it annihilates. The spectral theorem then writes any self-adjoint observable as a "menu" of its eigenvalues, each tagged with the projector onto the states that yield it, and those projectors tile the whole space with no overlap and no gaps.

Units check. Projectors and the identity are pure numbers (dimensionless), as they must be to be added to and equated with one another. In A = Σ_i λ_i P_i each eigenvalue λ_i carries the physical units of the observable A (energy for a Hamiltonian, action for spin), so both sides share those units; the dimensionless P_i merely distributes the weight. In Σ_i P_i = I both sides are dimensionless.

Limiting cases

Single eigenvalue (A = λI): one projector P₁ = I, so the resolution reduces to the identity itself — a degenerate observable that measures nothing.
Rank-one projector P = |ψ⟩⟨ψ| with ‖ψ‖ = 1: idempotency and self-adjointness are immediate, and P selects the component along |ψ⟩.
Two-outcome observable: A = λ₁P₁ + λ₂P₂ with P₂ = I − P₁, the standard yes/no or spin-up/spin-down structure.
Complete degeneracy within a level: P_i has rank > 1 and projects onto a multi-dimensional eigenspace; the eigenvalue is unchanged but the outcome does not fix the state.

Breaks when

Continuous spectrum. For position, momentum, or a free-particle Hamiltonian the eigenvalues form a continuum with no normalizable eigenvectors; the discrete sum Σ_i λ_iP_i is meaningless and must be replaced by the spectral integral ∫ λ dE(λ) over a projection-valued measure.
Non-normal / non-self-adjoint operators. A merely idempotent P² = P that is not self-adjoint is an oblique projector: it still splits the space but along a slanted complement, so ran P is no longer perpendicular to ker P and ‖Pv‖ ≤ ‖v‖ can fail. Defective (non-diagonalizable) operators admit no projector decomposition at all — they need the Jordan form.
Incomplete eigensystem. If the self-adjoint operator is only symmetric but not essentially self-adjoint (a domain subtlety on unbounded operators), the eigenvectors may fail to be complete and Σ_i P_i ≠ I.

Failure modes

Confusing idempotent with unitary. Students write P†P = I for a projector; the correct relations are P² = P and P† = P, and P is not invertible unless P = I.
Dropping self-adjointness and still assuming orthogonality. Assuming ran P ⊥ ker P from P² = P alone; orthogonality requires P† = P (Step 3), otherwise the projection is oblique.
Forgetting normalization in |ψ⟩⟨ψ|. Using an unnormalized |ψ⟩ gives P² = ‖ψ‖²P ≠ P, so it is not idempotent.
Summing probabilities that omit degeneracy. Writing ⟨ψ|P_i|ψ⟩ = |⟨e_i|ψ⟩|² for a degenerate level; the correct Born weight sums Σ_k |⟨e_i,k|ψ⟩|² over the whole eigenspace.
Treating Σ_i P_i = I as optional. Omitting an eigenvalue (e.g. truncating a basis) silently violates completeness and produces probabilities that do not sum to one.

Discussion

The equivalence "self-adjoint idempotent = orthogonal projection" is what makes measurement geometrically honest. An orthogonal projector is the operator that maps a state to the closest point in a subspace (Step 4), so collapse onto an eigenspace is the minimal, distance-minimizing update consistent with the measured value. The self-adjointness condition is precisely the metric statement that this closest-point map is symmetric — it does not distort angles — whereas a bare idempotent (oblique projector) is a shear that keeps a subspace fixed but shadows it onto a tilted complement.

The resolution of the identity Σ_i P_i = I is the completeness relation dressed as an operator identity. Sandwiching it between a state, ⟨ψ|ψ⟩ = Σ_i ⟨ψ|P_i|ψ⟩ = Σ_i p_i, shows immediately that the Born probabilities p_i = ⟨ψ|P_i|ψ⟩ sum to one for a normalized state. This is where the chance thread is anchored: probability conservation is not an extra axiom but a direct corollary of the projectors tiling the space.

The functional calculus (Step 8) is the practical payoff. Because commuting projectors turn operator functions into ordinary functions of eigenvalues, the Schrödinger propagator becomes e^−iHt/ħ = Σ_i e^−iE_it/ħ P_i and the thermal density operator becomes ρ = Z⁻¹Σ_i e^−βE_i P_i. The hard analytic content — existence of the eigenbasis — is imported once from the spectral theorem; everything downstream is bookkeeping with orthogonal projectors.

At the rigorous end, the leap from finite-dimensional intuition to infinite dimensions is nontrivial: the sum Σ_i P_i converges only in the strong operator topology, not in norm, and for operators with continuous spectrum the eigenprojectors are replaced by a projection-valued measure E(Ω) assigning an orthogonal projector to each Borel set of spectral values, with A = ∫ λ\,dE(λ). The discrete case treated here is the special instance where E is supported on isolated atoms and P_i = E({λ_i}).

Common misconceptions. A projector is not a probability — it is an operator whose expectation in a state is a probability. And "orthogonal projector" refers to orthogonality of the geometry (ran ⊥ ker), not to the projector being an orthogonal matrix; orthogonal projectors other than 0 and I are never invertible and so are never orthogonal/unitary matrices.

Worked examples

Spin-½ along z. S_z = (ħ/2)σ_z = (ħ/2) diag(1, −1) on basis {|↑⟩, |↓⟩}

Symbolic setup: the observable has eigenvalues λ₊ = +ħ/2 and λ₋ = −ħ/2 with eigenvectors |↑⟩, |↓⟩. A

P₊ = |↑⟩⟨↑| = [[1, 0],[0, 0]] , P₋ = |↓⟩⟨↓| = [[0, 0],[0, 1]]

Rank-one eigenprojectors. Check: P₊ + P₋ = [[1,0],[0,1]] = I ✓ and P₊P₋ = 0 ✓, and (ħ/2)P₊ + (−ħ/2)P₋ = (ħ/2)diag(1,−1) = S_z ✓. B

|ψ⟩ = α|↑⟩ + β|↓⟩ , p₊ = ⟨ψ|P₊|ψ⟩ = |α|² , ⟨S_z⟩ = (ħ/2)(|α|² − |β|²)

Born rule from the projector; expectation from the spectral resolution. B

α = 1/√5 , β = 2/√5 ⇒ |α|² = 0.20 , |β|² = 0.80 ⇒ ⟨S_z⟩ = (ħ/2)(0.20 − 0.80) = −0.30 ħ

Numbers in only now; check normalization |α|² + |β|² = 1 ✓. With ħ = 1.055×10⁻³⁴ J⋅s. A

p₊ = 0.20 , ⟨S_z⟩ = −0.30 ħ = −3.16×10⁻³⁵ J⋅s

Reading. The projector P₊ assigns a 20% chance of measuring spin-up; the resolution S_z = (ħ/2)P₊ − (ħ/2)P₋ gives an expectation tilted toward spin-down. Units: p₊ dimensionless; ⟨S_z⟩ in J⋅s (action), correct for spin.

Real symmetric 2×2. A = [[3, 1],[1, 3]]

Self-adjoint (symmetric, real). Characteristic equation (3−λ)² − 1 = 0 ⇒ λ = 4 or λ = 2. A

e₁ = (1, 1)/√2 (λ₁=4) , e₂ = (1, −1)/√2 (λ₂=2)

Orthonormal eigenvectors, as guaranteed for a self-adjoint operator with distinct eigenvalues. B

P₁ = e₁e₁^⊤ = ½[[1, 1],[1, 1]] , P₂ = e₂e₂^⊤ = ½[[1, −1],[−1, 1]]

Outer products form the eigenprojectors. Check P₁ + P₂ = [[1,0],[0,1]] = I ✓, P₁P₂ = 0 ✓. B

4P₁ + 2P₂ = [[2, 2],[2, 2]] + [[1, −1],[−1, 1]] = [[3, 1],[1, 3]] = A ✓

Spectral resolution verified numerically. A

√A = √4 P₁ + √2 P₂ = 2P₁ + 1.414 P₂ ≈ [[1.707, 0.293],[0.293, 1.707]]

Functional calculus f(A) = Σ f(λ_i)P_i with f = √ ⋅ ; verify (√A)² ≈ [[3.0, 1.0],[1.0, 3.0]] = A ✓. C

A = 4P₁ + 2P₂ , √A = 2P₁ + √2 P₂ ≈ [[1.707, 0.293],[0.293, 1.707]]

Reading. The matrix is diagonal in the projector basis; taking a square root is done outcome-by-outcome on the eigenvalues, exactly what the functional calculus promises. Units: the entries of A and its eigenvalues share whatever unit A carries; P_i dimensionless, so √A carries the square root of that unit.

Problems

Show that if P² = P and P† = P then I − P is also an orthogonal projector, and identify its range and kernel.
Solution
Idempotent: (I−P)² = I − 2P + P² = I − 2P + P = I − P. Self-adjoint: (I−P)† = I† − P† = I − P. Hence it is an orthogonal projector. Its range is ker P (since (I−P)v = 0 ⇔ v = Pv ⇔ v ∈ ran P, so ker(I−P) = ran P and by complementarity ran(I−P) = ker P).
A qubit is prepared in |ψ⟩ = cos(θ/2)|↑⟩ + sin(θ/2)|↓⟩ with θ = 60°. Compute the probability of measuring spin-up along z and the expectation ⟨S_z⟩.
Solution
p₊ = ⟨ψ|P₊|ψ⟩ = cos²(θ/2) = cos²(30°) = (√3/2)² = 0.75. Then p₋ = sin²(30°) = 0.25. Expectation ⟨S_z⟩ = (ħ/2)(0.75 − 0.25) = (ħ/2)(0.50) = 0.25 ħ = 2.64×10⁻³⁵ J⋅s.
For A = [[2, 0],[0, 5]], write down the two eigenprojectors, verify the resolution of the identity, and compute e^A via the functional calculus.
Solution
Eigenvalues 2, 5 with P₁ = [[1,0],[0,0]], P₂ = [[0,0],[0,1]]. Check P₁ + P₂ = I ✓, 2P₁ + 5P₂ = A ✓. Functional calculus: e^A = e²P₁ + e⁵P₂ = [[e², 0],[0, e⁵]] = [[7.389, 0],[0, 148.41]].
Give an explicit 2×2 real matrix that is idempotent (P² = P) but not self-adjoint, and show geometrically that its range and kernel are not orthogonal.
Solution
Take P = [[1, 1],[0, 0]]. Then P² = [[1,1],[0,0]] = P ✓, but P† = P^⊤ = [[1,0],[1,0]] ≠ P. Range is spanned by (1,0); kernel solves x + y = 0, spanned by (1,−1). Inner product (1,0)⋅(1,−1) = 1 ≠ 0, so they are oblique — this is an oblique (non-orthogonal) projector along the line y = −x onto the x-axis.
A self-adjoint operator has spectrum {1, 1, 4} (eigenvalue 1 doubly degenerate) on ℝ³ with the degenerate eigenspace spanned by e₁=(1,0,0), e₂=(0,1,0) and the λ=4 eigenvector e₃=(0,0,1). Write the spectral resolution and compute the probability of outcome λ=1 for the state |ψ⟩ = (1/√3)(1,1,1).
Solution
Degenerate projector P₁ = e₁e₁^⊤ + e₂e₂^⊤ = diag(1,1,0); P₂ = diag(0,0,1). Resolution A = 1⋅P₁ + 4⋅P₂, and P₁ + P₂ = I ✓. Born weight sums over the whole degenerate eigenspace: p(λ=1) = ⟨ψ|P₁|ψ⟩ = |⟨e₁|ψ⟩|² + |⟨e₂|ψ⟩|² = (1/√3)² + (1/√3)² = 1/3 + 1/3 = 2/3 ≈ 0.667. (And p(λ=4) = 1/3, summing to 1 ✓.)