Unitary Operators as Inner-Product Isometries

Derivation

Unitary Operators as Inner-Product Isometries

D-083 Home PU-104 Threads symmetry Depends on Existence and Uniqueness of the Adjoint, Cauchy–Schwarz and the Triangle Inequality

Statement

Let U be a bounded linear operator on a complex Hilbert space H with inner product ⟨·,·⟩ (antilinear in the first slot by our convention, linear in the second). Then U preserves the inner product, ⟨Ux,Uy⟩ = ⟨x,y⟩ for all x,y ∈ H, if and only if U†U = I. When U is additionally surjective (automatic in finite dimensions) this makes U unitary, U†U = UU† = I. Every eigenvalue of such a U then satisfies |λ| = 1: the spectrum lies on the unit circle.

Why it matters

Inner products encode every physically measurable quantity in quantum mechanics: probabilities |⟨φ|ψ⟩|², expectation values, and transition amplitudes. An operation that leaves all inner products fixed is exactly one that preserves probability, i.e. a legitimate symmetry or time evolution. This theorem is the algebraic content of "quantum evolution is unitary."

The eigenvalue statement |λ| = 1 is why phases, not decays, are the signature of conservative dynamics. A generator H = H† exponentiates to U = e^−iHt/ℏ whose eigenvalues e^{−iE t/ℏ} ride the unit circle for all time; nothing grows or shrinks. The same result underwrites the change-of-basis matrices of quantum information and the S-matrix of scattering theory.

Assumptions

The space is a complex inner-product (Hilbert) space.Over a real space the polarization identity has a different form and "unitary" becomes "orthogonal"; the eigenvalue argument still gives |λ| = 1 but complex eigenvalues need the complexification to be visible. The inner product is nondegenerate.If ⟨x,z⟩ = 0 for all x did not force z = 0, then ⟨x,(U†U−I)y⟩ = 0 ∀x would no longer imply U†U = I; the "only if" direction collapses. The adjoint U† exists and is unique.Guaranteed for bounded U by the Riesz representation theorem (prior result: adjoint-operator-existence-uniqueness). Drop boundedness and U† may be defined only on a dense domain, so U†U = I must be read carefully on domains — the finite-dimensional statement is unaffected. U is surjective (for the full unitary conclusion).Without surjectivity U†U = I still holds — U is an isometry — but UU† ≠ I is possible (e.g. the unilateral shift). Then U is not invertible and the spectrum can fill the closed unit disk rather than sit on the circle.

Derivation

⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y ∈ H

Hypothesis: U preserves the inner product. We derive the operator identity it forces. A

⟨Ux, Uy⟩ = ⟨x, U†U y⟩

Definition of the adjoint applied twice: ⟨Ux, w⟩ = ⟨x, U†w⟩ with w = Uy (prior result: adjoint existence and uniqueness). A

⟨x, U†U y⟩ = ⟨x, y⟩ ⟹ ⟨x, (U†U − I) y⟩ = 0

Equate steps 1 and 2, then use linearity in the second slot to collect terms. Holds for every x. A

(U†U − I) y = 0 for all y ⟹ U†U = I

Nondegeneracy: if ⟨x, z⟩ = 0 for all x then z = 0. Take x = (U†U − I)y to get ‖(U†U − I)y‖² = 0. B

U†U = I ⟹ ⟨Ux, Uy⟩ = ⟨x, U†U y⟩ = ⟨x, I y⟩ = ⟨x, y⟩

Converse direction: substitute the identity back. Each equality is definition of adjoint, then U†U = I. The "iff" is complete. A

Set y = x : ‖Ux‖² = ⟨Ux, Ux⟩ = ⟨x, x⟩ = ‖x‖²

Specialise preservation to equal arguments — U is a norm-preserving isometry. A

U ψ = λ ψ, ψ ≠ 0 ⟹ ‖ψ‖² = ‖Uψ‖² = ‖λψ‖² = |λ|² ‖ψ‖²

Feed an eigenvector into step 6 and pull the scalar out of the norm: ‖λψ‖ = |λ| ‖ψ‖. A

(|λ|² − 1) ‖ψ‖² = 0, ‖ψ‖ ≠ 0 ⟹ |λ| = 1

Divide by ‖ψ‖² > 0 (eigenvector nonzero). The eigenvalue sits on the unit circle. A

‖Ux‖ = ‖x‖ ∀x ⟹ ⟨Ux,Uy⟩ = ⟨x,y⟩ (polarization)

Sharper converse: norm preservation alone forces inner-product preservation, via 4⟨x,y⟩ = ‖x+y‖² − ‖x−y‖² − i‖x+iy‖² + i‖x−iy‖² applied to U-images. So "isometry" and "inner-product-preserving" coincide on a complex space. C

Result

⟨Ux, Uy⟩ = ⟨x, y⟩ ∀x,y ⟺ U†U = I (and then |λ| = 1 for every eigenvalue λ)

Reading. An operator leaves all inner products — hence all lengths, angles, and quantum probabilities — untouched exactly when its adjoint is its left inverse. On a finite-dimensional (or surjective) space this is full unitarity, U†U = UU† = I, and the whole spectrum is forced onto the unit circle in the complex plane: eigenvalues are pure phases e^iθ.

Units check. The inner product carries whatever units the state vectors do; U, U†, and I are all dimensionless linear maps, so U†U = I balances dimensionlessly. Eigenvalues λ = e^iθ are dimensionless with θ in radians. In the QM realisation U = e^−iHt/ℏ, the exponent Ht/ℏ has units (J·s)/(J·s) = dimensionless, as required.

Limiting cases

U = I: trivially preserves inner products; single eigenvalue λ = 1 = e^i·0 on the circle.
U = e^iφI (global phase): preserves all inner products, every eigenvalue is e^iφ — physically undetectable, the origin of the "global phase is unobservable" rule.
Real orthogonal O (OᵀO = I): the real special case; eigenvalues come in conjugate pairs e^±iθ plus possible ±1.
Hermitian U = U† that is also unitary: forces U² = I, so λ ∈ {+1, −1} — the reflection/involution limit (e.g. Pauli matrices).
Near-identity U = I + iεA: unitarity to first order in ε demands A = A†, recovering "generators of unitaries are Hermitian."

Breaks when

Non-surjective isometry (infinite dimensions). The unilateral shift S(a₁,a₂,…) = (0,a₁,a₂,…) obeys S†S = I and preserves inner products, yet SS† ≠ I. It is not unitary, is not invertible, and has no eigenvalues at all while its spectrum fills the closed unit disk. "Preserves inner product" ⇒ isometry, but ⇏ unitary here.
Non-normalisable / continuous spectrum. For operators with purely continuous spectrum (e.g. multiplication by e^iθ(x) on L²) there are no genuine eigenvectors ψ ∈ H; step 7 has nothing to act on. The statement about eigenvalues must be replaced by "the spectrum lies on the unit circle."
Indefinite metric spaces. On a Krein/pseudo-Hilbert space (e.g. the Minkowski or Lorentzian inner product used in Gupta–Bleuler electrodynamics) the "adjoint" is defined against an indefinite η; preserving η-products gives U†ηU = η, and eigenvalues can leave the unit circle (they pair as λ, 1/λ̄). Nondegeneracy of a positive product was essential to step 4.
Antilinear operators. Time-reversal T preserves |⟨Tx,Ty⟩| = |⟨x,y⟩| but conjugates it: ⟨Tx,Ty⟩ = ⟨y,x⟩ = ⟨x,y⟩*. It is antiunitary, not unitary; the linearity assumed in step 3 fails, so this theorem does not apply.

Failure modes

Confusing isometry with unitarity. Writing "U†U = I ⇒ UU† = I" as if automatic. True in finite dimensions (a left inverse of a square matrix is a two-sided inverse) but false in general — the shift is the standard counterexample.
Slot/convention slip in the adjoint. Using ⟨Ux,Uy⟩ = ⟨U†Ux,y⟩ with the wrong slot, giving UU† instead of U†U. Fix the antilinear slot convention first and stay consistent.
Claiming eigenvalues are real. Confusing unitary with Hermitian: unitary eigenvalues lie on the unit circle (|λ|=1), Hermitian eigenvalues lie on the real axis. Only their intersection, λ = ±1, is both.
Dropping the nonzero-eigenvector condition. Cancelling ‖ψ‖² in step 8 without noting ψ ≠ 0; the zero "eigenvector" is excluded by definition, which is what licenses the division.
Assuming eigenvectors exist. Applying step 7 to an operator with no point spectrum (continuous-spectrum unitaries) and concluding falsely that it has unit-modulus eigenvalues.
Global-phase double counting. Treating U and e^iφU as physically distinct: both preserve every |⟨φ|ψ⟩|², so the phase is unobservable.

Discussion

The theorem is the precise sense in which unitary operators are the "rigid motions" of Hilbert space. Just as an orthogonal matrix is exactly a length-and-angle-preserving map of Euclidean space, a unitary operator is exactly a map that preserves the Hermitian inner product — and hence every probability amplitude built from it. Wigner's theorem sharpens this: any symmetry preserving all transition probabilities |⟨φ|ψ⟩|² is realised by an operator that is either unitary or antiunitary. The present result handles the linear half; time reversal supplies the antilinear half.

The eigenvalue conclusion is the structural reason quantum evolution neither amplifies nor damps. Writing U(t) = e^−iHt/ℏ with H Hermitian, its eigenvalues are e^{−iE t/ℏ} — pure rotation of each energy eigenstate's phase at angular rate E/ℏ. Probability is conserved because |e^−iEt/ℏ|=1. The moment you want decay you must leave the unitary world (open systems, non-Hermitian effective Hamiltonians), and the eigenvalues then move off the unit circle into |λ|<1.

Spectrally, a unitary operator is unitarily diagonalisable (it is normal, UU†=U†U), so U = Σ_k e^iθ_k P_k with orthogonal projectors P_k. Distinct eigenvalues have orthogonal eigenspaces, exactly as for Hermitian operators — the only change is that the eigenvalues live on the circle instead of the line. This is the finite-dimensional shadow of the spectral theorem for unitaries, U = ∫_|z|=1 z dE(z).

In infinite dimensions the neat picture fractures instructively. Being an isometry (U†U=I) is strictly weaker than being unitary; the extra condition UU†=I is surjectivity, which cannot be dropped. The unilateral shift is the canonical isometry-but-not-unitary, with empty point spectrum and spectrum equal to the whole closed disk — a vivid reminder that "preserves the inner product" guarantees unit-modulus behaviour only where genuine eigenvectors exist, and that the clean spectrum-on-the-circle statement is really a theorem about unitaries, i.e. invertible isometries. The proper general statement uses the spectral measure supported on |z|=1.

Common misconceptions. "Unitary means the matrix has determinant 1" — no, that is special unitary SU(n); unitarity only forces |det U| = 1. "Unitary and Hermitian are the same because both are nice" — they are different constraints (U†U=I vs U=U†) with different spectra (circle vs line), overlapping only at eigenvalues ±1. "A norm-preserving map need not preserve inner products" — false on a complex space, where polarization recovers the full inner product from the norm (step 9).

Worked examples

Real rotation R(θ) = [[cosθ, −sinθ], [sinθ, cosθ]] — verify isometry and locate eigenvalues, θ = 30°.

A concrete orthogonal (real unitary) operator on ℝ² (or ℂ²). A

RᵀR = [[c,s],[−s,c]][[c,−s],[s,c]] = [[c²+s², 0],[0, c²+s²]] = I

Compute the product symbolically with c=cosθ, s=sinθ; c²+s²=1. So RᵀR=I: inner products preserved. A

det(R − λI) = (c−λ)² + s² = 0 ⟹ λ = c ± i s = cosθ ± i sinθ = e^±iθ

Characteristic polynomial; roots are pure phases. A

θ = 30° = π/6: λ = e^±iπ/6 = cos30° ± i sin30° = 0.8660 ± 0.5000 i

Insert the number last. Check |λ| = √(0.8660² + 0.5000²) = √(0.75+0.25) = 1. A

RᵀR = I, λ = e^±iπ/6 = 0.866 ± 0.500 i, |λ| = 1

Reading. A 30° rotation preserves all lengths and angles; its eigenvalues sit on the unit circle at ±30°, as the theorem demands. Dimensionless throughout.

Phase gate on a qubit: U = [[1, 0], [0, e^iα]], α = π/4. Verify U†U = I and eigenvalues; evolve |ψ⟩ = (|0⟩ + |1⟩)/√2 and check the norm.

A genuinely complex 2×2 unitary from quantum information. A

U† = [[1,0],[0, e^−iα]], U†U = [[1·1, 0],[0, e^−iαe^iα]] = [[1,0],[0,1]] = I

Adjoint = conjugate transpose; e^−iαe^iα=1. Inner products preserved. A

Diagonal ⟹ eigenvalues are the diagonal entries: λ₁ = 1 = e^i·0, λ₂ = e^iα, both with |λ| = 1

Eigenvalues read off; both on the unit circle for any real α. A

U|ψ⟩ = (|0⟩ + e^iα|1⟩)/√2, ‖U|ψ⟩‖² = (|1|² + |e^iα|²)/2 = (1+1)/2 = 1 = ‖ψ‖²

Norm from squared amplitudes; the phase drops out of the modulus. A

α = π/4: e^iα = cos45° + i sin45° = 0.7071 + 0.7071 i, U|ψ⟩ = (|0⟩ + (0.7071+0.7071i)|1⟩)/√2

Number inserted at the end; state stays normalised, only the relative phase advanced by 45°. A

U†U = I, λ = {1, e^iπ/4}, ‖U|ψ⟩‖ = ‖|ψ⟩‖ = 1

Reading. The phase gate leaves total probability at 1 while rotating the relative phase of |1⟩ by 45°. Its eigenvalues 1 and e^iπ/4 lie on the unit circle — probability is conserved, phases evolve. All quantities dimensionless.

Problems

Show that the Pauli matrix X = [[0,1],[1,0]] is both Hermitian and unitary, and find its eigenvalues. What does the theorem predict about where they must lie, and how is that consistent with X being Hermitian?
Solution
X† = X (real symmetric), so Hermitian. X†X = X² = [[0,1],[1,0]][[0,1],[1,0]] = [[1,0],[0,1]] = I, so unitary. Eigenvalues: det(X−λI) = λ²−1 = 0 ⟹ λ = ±1. Unitarity forces |λ|=1 (unit circle); Hermiticity forces λ real (real axis). The intersection is λ = ±1 — exactly what we get. Any operator that is both unitary and Hermitian is an involution, X² = I.
An operator on ℂ² is U = (1/√2)[[1, 1],[1, −1]] (the Hadamard gate). Verify U†U = I and compute its eigenvalues.
Solution
U† = U (real symmetric). U²= (1/2)[[1,1],[1,−1]][[1,1],[1,−1]] = (1/2)[[2,0],[0,2]] = I, so U†U = U² = I: unitary (and Hermitian). Eigenvalues: det(U−λI)=0. Trace = 0, det = (1/2)(−1)−(1/2)(1) = −1, so λ² − (tr)λ + det = λ² − 0 − 1 = 0 ⟹ λ = ±1. Both on the unit circle, and real because U is also Hermitian.
Let U = e^iθI on ℂⁿ. Prove it preserves the inner product and give its eigenvalues with multiplicity. Explain the physical meaning of the fact that U and I produce identical measurement statistics.
Solution
U†U = e^−iθe^iθI = I, so it is unitary. ⟨Ux,Uy⟩ = e^−iθe^iθ⟨x,y⟩ = ⟨x,y⟩. The only eigenvalue is e^iθ with multiplicity n (every vector is an eigenvector). Physically U multiplies the whole state by a global phase; since observables depend on |⟨φ|ψ⟩|² and |e^iθ|²=1, the phase cancels in every probability. Hence global phase is unobservable — states are rays, not vectors.
The unilateral shift S on ℓ² acts by S(a₁,a₂,a₃,…) = (0,a₁,a₂,…). Show S†S = I (so S preserves inner products) but SS† ≠ I, and explain why the eigenvalue conclusion of the theorem does not apply.
Solution
The adjoint is the backward shift S†(b₁,b₂,b₃,…) = (b₂,b₃,…). Then S†S(a₁,a₂,…) = S†(0,a₁,a₂,…) = (a₁,a₂,…), so S†S = I — S is an isometry and preserves all inner products. But SS†(a₁,a₂,…) = S(a₂,a₃,…) = (0,a₂,a₃,…) ≠ (a₁,a₂,…) whenever a₁ ≠ 0, so SS† ≠ I: S is not surjective, not invertible, not unitary. Suppose Sψ = λψ. Comparing components: the first gives 0 = λa₁, and the (k+1)-th gives a_k = λ a_{k+1}. If λ=0 then all a_k=0; if λ≠0 then a₁=0 forces a₂=0, then all a_k=0. So S has no eigenvectors. Step 7 needs a nonzero eigenvector to conclude |λ|=1; with none, the eigenvalue statement is vacuous. (Its spectrum is the whole closed unit disk.) This is why the theorem's spectrum-on-the-circle claim requires unitarity, not mere isometry.
A Hamiltonian has energies E₀ = 0 and E₁ = 2.00 eV. Write the eigenvalues of the evolution operator U(t) = e^−iHt/ℏ at t = 1.03 fs, verify they lie on the unit circle, and state the relative phase accumulated.
Solution
Eigenvalues of U(t) are e^{−iE_k t/ℏ}. For E₀=0: λ₀ = e⁰ = 1. For E₁: phase φ = E₁ t/ℏ. Symbolically λ₁ = e^−iφ with |λ₁| = 1 automatically (real φ). Numbers: E₁ = 2.00 eV = 2.00 × 1.602×10⁻¹⁹ J = 3.204×10⁻¹⁹ J; ℏ = 1.055×10⁻³⁴ J·s; t = 1.03×10⁻¹⁵ s. Then φ = (3.204×10⁻¹⁹)(1.03×10⁻¹⁵)/(1.055×10⁻³⁴) = (3.300×10⁻³⁴)/(1.055×10⁻³⁴) ≈ 3.128 rad ≈ π. So λ₁ = e^−i·3.128 ≈ −1.000 (i.e. cos3.128 + i sin3.128 ≈ −0.99991 + 0.0136 i), modulus 1. The relative phase between the two levels is φ ≈ 3.13 rad ≈ π: after ~1.03 fs the excited state has picked up essentially a half-cycle (sign flip) relative to the ground state. Both eigenvalues sit on the unit circle, confirming probability conservation.