Source: Friedberg, Insel, Spence -- Linear Algebra 4th ed., Ch. 6.5-6.6
Tags: unitary operator, orthogonal operator, unitary matrix, orthogonal matrix, isometry, length-preserving, spectral theorem, spectral decomposition, orthogonal projection, rigid motion, rotation, reflection, conic sections, principal axis theorem, unitarily equivalent, MATH 301, UIUC, abstract linear algebra
Difficulty: Advanced Prerequisites: Inner products, norms, orthonormal bases (6.1-6.2). Adjoint, normal, and self-adjoint operators (6.3-6.4). Eigenvalues and diagonalisation (Chapter 5).
This is the culmination of Chapter 6. Section 6.5 studies the operators that preserve length (and therefore preserve the entire inner product structure): unitary operators over C and orthogonal operators over R. These are the inner-product-space analogue of invertible linear maps, and their matrices are the unitary and orthogonal matrices familiar from applied mathematics. Section 6.6 delivers the Spectral Theorem, the single most important result in the chapter. It says that any normal operator (over C) or self-adjoint operator (over R) can be decomposed into a weighted sum of orthogonal projections onto its eigenspaces. This decomposition is the conceptual endpoint of the course's treatment of inner product spaces.
Unitary and orthogonal operators are those that preserve lengths and inner products; their matrices satisfy A*A = I. The Spectral Theorem says every normal (complex) or self-adjoint (real) operator T on a finite-dimensional inner product space can be written as T = λ₁T₁ + λ₂T₂ + ... + λₖTₖ, where the λᵢ are the distinct eigenvalues and each Tᵢ is the orthogonal projection onto the corresponding eigenspace.
Unitary operator (F = C)
A linear operator T on a finite-dimensional complex inner product space such that ‖T(x)‖ = ‖x‖ for all x. Think of it as a transformation that preserves all distances and angles in a complex vector space.
Orthogonal operator (F = R)
The real analogue: a linear operator T on a finite-dimensional real inner product space such that ‖T(x)‖ = ‖x‖ for all x. Rotations and reflections in R² are the prototypical examples.
Unitary matrix
A square matrix A satisfying A*A = AA* = I. Equivalently, the columns (or rows) of A form an orthonormal basis for F^n.
Orthogonal matrix
A real square matrix A satisfying AᵗA = AAᵗ = I. The real version of a unitary matrix.
Unitarily equivalent matrices
A and B are unitarily equivalent if B = P*AP for some unitary matrix P. This is the inner-product-space version of similarity. In simple terms, the matrices represent the same operator with respect to different orthonormal bases.
Orthogonally equivalent matrices
The real version: B = PᵗAP for some orthogonal matrix P.
Orthogonal projection
A projection T such that R(T)⊥ = N(T). Equivalently, T² = T = T*. The orthogonal projection onto a subspace W is the unique such operator with range W.
Spectral decomposition
The expression T = λ₁T₁ + λ₂T₂ + ... + λₖTₖ, where λ₁, ..., λₖ are the distinct eigenvalues of T and each Tᵢ is the orthogonal projection onto the eigenspace Wᵢ.
Resolution of the identity
The decomposition I = T₁ + T₂ + ... + Tₖ from the Spectral Theorem.
Spectrum
The set {λ₁, λ₂, ..., λₖ} of distinct eigenvalues of T.
Rigid motion
A function f : V → V on a real inner product space preserving distances: ‖f(x) - f(y)‖ = ‖x - y‖ for all x, y.
The following are all equivalent for a linear operator T on a finite-dimensional inner product space:
(a) TT* = T*T = I
(b) ⟨T(x), T(y)⟩ = ⟨x, y⟩ for all x, y (T preserves the inner product)
(c) T sends every orthonormal basis to an orthonormal basis
(d) T sends some orthonormal basis to an orthonormal basis
(e) ‖T(x)‖ = ‖x‖ for all x (T preserves norms)
The fact that (e) implies (a) is perhaps the most surprising: preserving lengths automatically preserves angles, inner products, and orthonormality.
From (a), unitary and orthogonal operators are automatically normal, so all the results about normal operators apply.
Every eigenvalue of a unitary or orthogonal operator has absolute value 1.
Corollary 1 (real): V has an orthonormal basis of eigenvectors of T with eigenvalues of absolute value 1 if and only if T is both self-adjoint and orthogonal. (Over R, the only eigenvalues of absolute value 1 are ±1.)
Corollary 2 (complex): V has an orthonormal basis of eigenvectors of T with eigenvalues of absolute value 1 if and only if T is unitary.
A matrix A is unitary (orthogonal) if and only if its columns form an orthonormal basis for F^n, which happens if and only if its rows do.
T is unitary [orthogonal] if and only if [T]β is unitary [orthogonal] for some (equivalently, every) orthonormal basis β.
Theorem 6.19: A complex matrix is normal if and only if it is unitarily equivalent to a diagonal matrix.
Theorem 6.20: A real matrix is symmetric if and only if it is orthogonally equivalent to a real diagonal matrix.
Theorem 6.21 (Schur, matrix form): every complex matrix is unitarily equivalent to an upper triangular matrix. Every real matrix whose characteristic polynomial splits is orthogonally equivalent to a real upper triangular matrix.
Every orthogonal operator on R² is either:
A rotation (det = 1), with matrix [[cos θ, -sin θ], [sin θ, cos θ]]
A reflection about a line through the origin (det = -1), with matrix [[cos 2α, sin 2α], [sin 2α, -cos 2α]], where α is the angle from the positive x-axis to the line
Every rigid motion on a finite-dimensional real inner product space can be written uniquely as an orthogonal operator followed by a translation: f = g ∘ T, where T is orthogonal and g is translation by f(0).
On R², this means every rigid motion is either a rotation-then-translate or a reflection-then-translate.
A quadratic equation ax² + 2bxy + cy² + dx + ey + f = 0 can be simplified by choosing coordinates along the eigenvectors of the symmetric matrix A = [[a, b], [b, c]]. The orthogonal matrix P of normalised eigenvectors diagonalises A, and the substitution X = PX' eliminates the cross-term, yielding λ₁(x')² + λ₂(y')² plus lower-order terms. The eigenvalues λ₁, λ₂ are the coefficients of the squared terms in the new coordinate system.
A linear operator T is an orthogonal projection if and only if T² = T = T*. This algebraic characterisation is the key to the Spectral Theorem.
Given an orthonormal basis β adapted to the decomposition V = W ⊕ W⊥, the orthogonal projection onto W has matrix [Iₖ, 0; 0, 0], where k = dim(W).
Let T be a linear operator on a finite-dimensional inner product space V over F, with distinct eigenvalues λ₁, ..., λₖ. Assume T is normal (if F = C) or self-adjoint (if F = R). Let Wᵢ be the eigenspace for λᵢ and Tᵢ the orthogonal projection of V onto Wᵢ. Then:
(a) V = W₁ ⊕ W₂ ⊕ ... ⊕ Wₖ
(b) Wᵢ⊥ = the direct sum of all Wⱼ for j ≠ i
(c) TᵢTⱼ = δᵢⱼTᵢ (the projections are mutually orthogonal)
(d) I = T₁ + T₂ + ... + Tₖ (resolution of the identity)
(e) T = λ₁T₁ + λ₂T₂ + ... + λₖTₖ (spectral decomposition)
Corollary 1: T is normal (over C) if and only if T* = g(T) for some polynomial g.
Corollary 2: T is unitary if and only if T is normal and every eigenvalue has absolute value 1.
Corollary 3: T is self-adjoint if and only if T is normal and every eigenvalue is real.
Corollary 4: each orthogonal projection Tⱼ in the spectral decomposition is itself a polynomial in T.
Polynomial functions of T: if T = Σ λᵢTᵢ is the spectral decomposition, then g(T) = Σ g(λᵢ)Tᵢ for any polynomial g. This is a powerful tool for computing matrix functions.
Rotation matrix on R²: [[cos θ, -sin θ], [sin θ, cos θ]], det = 1
Reflection matrix on R² about line at angle α: [[cos 2α, sin 2α], [sin 2α, -cos 2α]], det = -1
Spectral decomposition: T = λ₁T₁ + λ₂T₂ + ... + λₖTₖ
Resolution of the identity: I = T₁ + T₂ + ... + Tₖ
Polynomial in T: g(T) = Σ g(λᵢ)Tᵢ
Orthogonal projection characterisation: T² = T = T*
Unitary/orthogonal matrix condition: A*A = AA* = I
Unitary and orthogonal matrices are the foundation of coordinate transformations in physics and engineering. Rotations in 3D graphics, the discrete Fourier transform (a unitary matrix), and quantum mechanical evolution operators are all unitary. The Spectral Theorem underpins principal component analysis (PCA) in statistics, where a symmetric covariance matrix is diagonalised to find the directions of greatest variance. The principal axis theorem for conic sections is a direct geometric application of orthogonal diagonalisation.
Students frequently confuse "orthogonal matrix" (AᵗA = I) with "orthogonal operator" (preserves norms). They are the same thing seen from different angles: the matrix is orthogonal if and only if the operator it represents (with respect to an orthonormal basis) is orthogonal.
Not every orthogonal operator is diagonalisable over R. A rotation of R² by an angle other than 0 or π has no real eigenvectors and is not diagonalisable over R, even though it is orthogonal.
Students sometimes think the Spectral Theorem applies to all diagonalisable operators. It does not: it applies specifically to normal (C) or self-adjoint (R) operators, and the decomposition uses orthogonal projections, not arbitrary ones.
The spectral decomposition T = Σ λᵢTᵢ looks like it depends on an eigenvalue ordering, but it is unique up to reordering.
⚠️ Know the five equivalent conditions for unitary/orthogonal operators (Theorem 6.18). Exams may ask you to show one condition implies another.
⚠️ Be able to find the orthogonal matrix P that diagonalises a real symmetric matrix A (i.e. find eigenvalues, orthonormal eigenvectors, form P, verify PᵗAP = D).
⚠️ The Spectral Theorem statement and its parts (a)-(e) are likely to appear. Know what the spectral decomposition means and how to construct it.
⚠️ Be able to write down the orthogonal projection Tᵢ onto an eigenspace Wᵢ given an orthonormal basis for Wᵢ, and verify the resolution of the identity.
⚠️ The classification of orthogonal operators on R² (rotation vs reflection, determined by the sign of the determinant) is a common short-answer or conceptual question.
⚠️ The principal axis theorem (eliminating the cross-term in a quadratic equation by orthogonal change of coordinates) is a classic application question.
True or False: Every unitary matrix is normal. True. TT* = I = T*T.
True or False: The product of two orthogonal matrices is orthogonal. True.
True or False: If T is an orthogonal projection, then T is self-adjoint. True. T² = T = T* characterises orthogonal projections.
Fill in the blank: In the spectral decomposition T = Σ λᵢTᵢ, the Tᵢ satisfy TᵢTⱼ = ____ for i ≠ j. 0 (the zero operator)
Fill in the blank: A complex n × n matrix is normal if and only if it is unitarily equivalent to a ____ matrix. Diagonal
Q: Let A = [[4, 2, 2], [2, 4, 2], [2, 2, 4]]. Find an orthogonal matrix P and diagonal matrix D such that PᵗAP = D.
A: The eigenvalues are 2 (multiplicity 2) and 8 (multiplicity 1). An orthonormal basis of eigenvectors: for λ = 2, use Gram-Schmidt on {(-1,1,0), (-1,0,1)} to get {(1/√2)(-1,1,0), (1/√6)(1,1,-2)}. For λ = 8: (1/√3)(1,1,1). Forming P from these columns gives PᵗAP = diag(2, 2, 8).
Q: State the Spectral Theorem. What are the hypotheses and the conclusion?
A: Let V be a finite-dimensional inner product space over F, and T a linear operator on V. Assume T is normal if F = C, or self-adjoint if F = R. Let λ₁, ..., λₖ be the distinct eigenvalues, Wᵢ the eigenspace for λᵢ, and Tᵢ the orthogonal projection onto Wᵢ. Then V = W₁ ⊕ ... ⊕ Wₖ, the projections satisfy TᵢTⱼ = δᵢⱼTᵢ, I = T₁ + ... + Tₖ, and T = λ₁T₁ + ... + λₖTₖ.
Q: Let T be a normal operator on a finite-dimensional complex inner product space. Prove that T is unitary if and only if every eigenvalue has absolute value 1.
A: (⇒) If T is unitary and T(x) = λx with x ≠ 0, then ‖x‖ = ‖T(x)‖ = |λ|‖x‖, so |λ| = 1. (⇐) Let T = Σ λᵢTᵢ be the spectral decomposition. If |λᵢ| = 1 for all i, then TT* = Σ |λᵢ|²Tᵢ = Σ Tᵢ = I.
Q: Show that an orthogonal operator on R² with determinant -1 is a reflection.
A: T(e₁) = (cos θ, sin θ) for some θ. Since T preserves orthonormality and det(A) = -1, T(e₂) must be (sin θ, -cos θ), giving A = [[cos θ, sin θ], [sin θ, -cos θ]]. Writing α = θ/2, this is [[cos 2α, sin 2α], [sin 2α, -cos 2α]], the reflection about the line at angle α to the x-axis.
Q: Use the principal axis theorem to eliminate the cross-term in 2x² - 4xy + 5y² = 36.
A: The associated matrix is A = [[2, -2], [-2, 5]]. Eigenvalues: 1 and 6. Orthonormal eigenvectors: (1/√5)(2, 1) for λ = 1, (1/√5)(-1, 2) for λ = 6. Set P with these columns. Then X = PX' transforms the equation to (x')² + 6(y')² = 36, an ellipse.
The Spectral Theorem is the culmination of the entire inner product spaces chapter. It connects backward to diagonalisation (Chapter 5), which handled arbitrary vector spaces, and forward to the Singular Value Decomposition (Section 6.7), which extends spectral ideas to non-square and non-normal matrices. In functional analysis, the Spectral Theorem generalises to infinite-dimensional Hilbert spaces, where it is the foundation of quantum mechanics. The classification of rigid motions (Theorem 6.22) connects to symmetry groups studied in abstract algebra.
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