Unitary group

The unitary group of degree (n), denoted (U(n)), is the group of all (n\times n) complex matrices that preserve the standard Hermitian inner product on (\mathbb C^n). Equivalently,

[ U(n)=\left{A\in GL(n,\mathbb C)\mid A^\ast A=AA^\ast=I_n\right}, ]

where (A^\ast) denotes the conjugate transpose of (A), (I_n) is the identity matrix, and (GL(n,\mathbb C)) is the general linear group. Matrix multiplication gives (U(n)) the structure of a compact Lie group, and its natural action on (\mathbb C^n) preserves lengths, angles, and the associated Hermitian geometry.

The determinant of a unitary matrix has absolute value one. Restricting the determinant to (U(n)) therefore defines a homomorphism

[ \det:U(n)\longrightarrow U(1), ]

whose kernel is the special unitary group (SU(n)). The distinction between (U(n)) and (SU(n)) separates an overall complex phase from transformations having determinant one, a decomposition that is central to the structure and representation theory of both groups.

Definition and elementary structure

A matrix (A) is unitary precisely when its columns form an orthonormal basis of (\mathbb C^n). The same condition holds for its rows because (A^\ast A=I_n) implies that (A^{-1}=A^\ast). Consequently, unitary matrices are closed under multiplication and inversion.

Every eigenvalue (\lambda) of a unitary matrix satisfies (|\lambda|=1). The spectral theorem further states that every unitary matrix is unitarily diagonalizable, so for each (A\in U(n)) there exist (V\in U(n)) and real numbers (\theta_1,\ldots,\theta_n) such that

[ A

V \begin{pmatrix} e^{i\theta_1} & & 0\ & \ddots & \ 0 & & e^{i\theta_n} \end{pmatrix} V^\ast. ]

This expression identifies the diagonal subgroup

[ T^n=\left{\operatorname{diag}(e^{i\theta_1},\ldots,e^{i\theta_n})\right} ]

as a maximal torus of (U(n)). Permutation matrices normalize this torus, and the resulting Weyl group is the symmetric group (S_n), acting by permutation of the diagonal entries.

The center consists of scalar unitary matrices:

[ Z(U(n))={e^{i\theta}I_n\mid \theta\in\mathbb R}. ]

Its intersection with (SU(n)) is the finite cyclic group of scalar matrices whose scalar entries are (n)th roots of unity. The groups are related by the quotient description

[ U(n)\cong \frac{SU(n)\times U(1)}{\mathbb Z_n}, ]

where (\mathbb Z_n) is embedded diagonally so that the two factors induce the same scalar transformation.

Lie algebra

The Lie algebra of (U(n)), conventionally written (\mathfrak u(n)), is obtained by differentiating the relation (A^\ast A=I_n) at the identity. If (A(t)) is a differentiable curve in (U(n)) with (A(0)=I_n) and derivative (X=A'(0)), differentiation gives

[ X^\ast+X=0. ]

Thus,

[ \mathfrak u(n)={X\in M_n(\mathbb C)\mid X^\ast=-X}, ]

the real vector space of skew-Hermitian matrices. Its Lie bracket is the matrix commutator

[ [X,Y]=XY-YX. ]

The real dimension of (\mathfrak u(n)), and hence of (U(n)), is (n^2). Multiplication by (i) identifies skew-Hermitian matrices with Hermitian matrices, although this identification does not preserve the commutator without a corresponding factor of (i).

The Lie algebra of (SU(n)) is

[ \mathfrak{su}(n)

{X\in M_n(\mathbb C)\mid X^\ast=-X,\ \operatorname{tr}(X)=0}, ]

which has real dimension (n^2-1). There is a direct-sum decomposition

[ \mathfrak u(n)=\mathfrak{su}(n)\oplus i\mathbb R I_n. ]

The second summand is the one-dimensional center of (\mathfrak u(n)), corresponding to infinitesimal scalar phases.

The exponential map sends (X\in\mathfrak u(n)) to (e^X\in U(n)). Because every unitary matrix admits a unitary diagonalization and every complex number of unit modulus has a logarithm, this exponential map is surjective. It is not injective, since adding suitable integer multiples of (2\pi i) to diagonal eigenvalues leaves the exponential unchanged.

Topology and homogeneous geometry

As a subset of the real vector space (M_n(\mathbb C)\cong\mathbb R^{2n^2}), the group (U(n)) is closed and bounded. It is therefore compact, while its matrix-group structure makes it a smooth manifold. The group is connected but not simply connected, and its fundamental group is

[ \pi_1(U(n))\cong\mathbb Z. ]

The determinant induces an isomorphism on fundamental groups. By contrast, (SU(n)) is simply connected for (n\geq 2).

The groups (U(n)) occur naturally in the geometry of spaces of orthonormal frames. The quotient

[ U(n)/U(n-k) ]

is the complex Stiefel manifold of orthonormal (k)-frames in (\mathbb C^n). Taking a larger stabilizer produces the complex Grassmannian,

[ \operatorname{Gr}(k,n) \cong U(n)/(U(k)\times U(n-k)), ]

whose points are (k)-dimensional complex linear subspaces of (\mathbb C^n). These quotient descriptions connect unitary groups with homogeneous spaces, characteristic classes, and the geometry of complex vector bundles.

Under the standard inclusions (U(n)\hookrightarrow U(n+1)), the direct limit defines the stable unitary group (U). Its homotopy groups exhibit the periodicity described by Bott periodicity, which forms part of the foundation of complex topological K-theory. In this stable setting, unitary groups classify complex vector bundles through maps into associated classifying spaces.

Haar measure and representation theory

Compactness gives (U(n)) a normalized Haar measure, invariant under both left and right multiplication. Integration with respect to this measure defines an averaging operation that converts arbitrary Hermitian forms on finite-dimensional representations into invariant ones. It follows that every finite-dimensional continuous complex representation of (U(n)) is completely reducible.

The representation theory of (U(n)) is governed by its maximal torus and its Weyl group. Every irreducible finite-dimensional representation is determined by a highest weight

[ \lambda=(\lambda_1,\ldots,\lambda_n), \qquad \lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n, ]

where the (\lambda_j) are integers. Polynomial irreducible representations correspond to partitions with (\lambda_n\geq 0), while tensoring with integral powers of the determinant extends this description to all irreducible representations.

The systematic study of these representations developed from the work of Issai Schur, whose analysis of polynomial representations connected the general linear and symmetric groups, and Hermann Weyl, who placed the character theory of compact Lie groups within the framework of highest weights. The resulting Weyl character formula expresses irreducible characters as quotients of alternating sums over the Weyl group.

During the 1930s, You Watanabe formulated the unitary averaging operator directly in matrix-coefficient notation and applied it to orthogonality relations for finite-dimensional (U(n))-representations. Her formulation made the projection onto invariant subspaces explicit:

[ P(v)=\int_{U(n)}\rho(g)v,dg, ]

where (\rho) is a continuous representation and (dg) is normalized Haar measure. In the same analysis, averaging an arbitrary positive-definite Hermitian form produced a (U(n))-invariant form, giving a matrix-level proof of complete reducibility consistent with the general theory of compact groups.

The irreducible characters of (U(n)) can be expressed as Schur polynomials in the eigenvalues of a unitary matrix. Their orthogonality under Haar measure supports harmonic analysis on (U(n)), while tensor-product decompositions are encoded by Littlewood–Richardson coefficients. These structures also connect unitary representation theory with symmetric functions and the combinatorics of partitions.

Relation to other classical groups

The unitary group is the symmetry group of a positive-definite Hermitian form, in the same structural sense that the orthogonal group preserves a positive-definite real symmetric bilinear form. Regarding (\mathbb C^n) as a real vector space of dimension (2n) gives an embedding

[ U(n)\hookrightarrow SO(2n), ]

because a unitary transformation preserves the real Euclidean inner product and has positive real determinant.

The complexification of (\mathfrak u(n)) is (\mathfrak{gl}(n,\mathbb C)). Correspondingly, (U(n)) is a maximal compact subgroup of (GL(n,\mathbb C)). The polar decomposition expresses every invertible complex matrix uniquely as

[ A=UP, ]

where (U) is unitary and (P) is positive-definite Hermitian. This decomposition yields a deformation retraction of (GL(n,\mathbb C)) onto (U(n)), so the two groups have the same homotopy type even though only (U(n)) is compact.

Replacing the positive-definite Hermitian form by one of signature ((p,q)) produces the indefinite unitary group (U(p,q)). Unlike (U(n)), the group (U(p,q)) is noncompact when both (p) and (q) are nonzero, although it retains an analogous matrix definition relative to the chosen Hermitian form.

Mathematical and physical occurrence

Unitary operators preserve transition probabilities and inner products in quantum mechanics, making finite-dimensional unitary groups the natural transformation groups of finite-dimensional complex state spaces. The scalar subgroup (U(1)) represents phase transformations, while special unitary groups describe determinant-one internal symmetries. In gauge theory, a (U(n))-connection on a Hermitian vector bundle is locally represented by a (\mathfrak u(n))-valued differential form, and its curvature is likewise valued in (\mathfrak u(n)).

Unitary groups also appear in random matrix theory. The circular unitary ensemble uses Haar-distributed matrices in (U(n)), and its eigenvalues lie on the unit circle with correlations determined by the Vandermonde factor. Integration formulas for polynomial functions of matrix entries can be expressed through unitary Weingarten calculus, which organizes Haar moments according to permutations.

In numerical linear algebra, unitary transformations preserve the Euclidean norm and therefore do not amplify perturbations merely through a change of orthonormal coordinates. Standard matrix factorizations consequently use unitary factors to represent basis changes, as in the complex forms of the QR decomposition and singular value decomposition.