Unitary matrix
A complex square matrix (U) is a unitary matrix when its conjugate transpose, denoted (U^{*}), is also its inverse. Equivalently,
[ U^{}U=UU^{}=I, ]
where (I) is the identity matrix of the same dimension. Unitary matrices are the complex analogues of orthogonal matrices, and they describe precisely the linear transformations of a finite-dimensional complex inner-product space that preserve its Hermitian inner product.
If (U=(u_{jk})) has columns (u_1,\ldots,u_n), then the relation (U^{*}U=I) is equivalent to
[ \langle u_j,u_k\rangle=\delta_{jk}. ]
Thus, the columns of a unitary matrix form an orthonormal basis of (\mathbb C^n). The rows satisfy the corresponding condition because a square matrix with (U^{}U=I) is invertible and consequently also obeys (UU^{}=I).
Algebraic properties
The unitary matrices of order (n) form the unitary group (U(n)) under matrix multiplication. Closure follows from
[ (UV)^{}(UV)=V^{}U^{}UV=V^{}V=I. ]
The identity matrix is unitary, and the inverse of a unitary matrix is its conjugate transpose. These relations make (U(n)) both a matrix group and a compact real Lie group of dimension (n^2).
Taking determinants in the defining equation gives
[ \overline{\det U}\det U=1, ]
so every unitary matrix has a determinant of absolute value one. The unitary matrices whose determinant equals one constitute the special unitary group (SU(n)), a closed normal subgroup of (U(n)) with real dimension (n^2-1).
Every entry of a unitary matrix has absolute value at most one. More generally, the sum of the squared absolute values in any row or column equals one. Individual entries need not have absolute value one, because the normalization condition applies to each entire row or column rather than to every component separately.
The product and inverse of unitary matrices remain unitary, whereas the sum of two unitary matrices is generally not unitary. Scalar multiplication preserves unitarity exactly when the scalar lies on the unit circle. Consequently, if (|\alpha|=1), then (\alpha U) is unitary whenever (U) is unitary.
Preservation of geometry
For vectors (x,y\in\mathbb C^n), a unitary matrix satisfies
[ \langle Ux,Uy\rangle =(Ux)^{}(Uy) =x^{}U^{}Uy =x^{}y =\langle x,y\rangle. ]
It therefore preserves the norm induced by the inner product:
[ |Ux|_2=|x|_2. ]
The same identity implies preservation of angles in the Hermitian sense and preservation of orthogonality. Conversely, every complex-linear transformation preserving the standard Hermitian inner product is represented by a unitary matrix relative to the standard basis.
A unitary change of basis leaves the matrix description of an operator altered by a unitary similarity,
[ A\longmapsto U^{*}AU. ]
Matrices related in this manner represent the same linear operator in different orthonormal bases. Unitary similarity preserves the characteristic polynomial, the eigenvalues with their algebraic multiplicities, the trace, the determinant, and the singular values.
Unitary transformations also preserve the Euclidean volume on (\mathbb R^{2n}) obtained by identifying (\mathbb C^n) with a real vector space. Their complex determinants may contain a phase factor, but the corresponding real determinant equals (|\det U|^2=1).
Spectral structure
Every eigenvalue of a unitary matrix lies on the unit circle. If (Uv=\lambda v) for a nonzero vector (v), norm preservation yields
[ |v|_2=|Uv|_2=|\lambda v|_2=|\lambda|,|v|_2, ]
and therefore (|\lambda|=1).
A unitary matrix is normal, since (U^{}U=UU^{}). The finite-dimensional spectral theorem consequently provides a unitary matrix (V) and real numbers (\theta_1,\ldots,\theta_n) such that
[ U
V \begin{pmatrix} e^{i\theta_1} & & 0\ & \ddots & \ 0 & & e^{i\theta_n} \end{pmatrix} V^{*}. ]
Hence every unitary matrix is unitarily diagonalizable, including those with repeated eigenvalues. Its eigenspaces associated with distinct eigenvalues are mutually orthogonal.
The diagonal form also supplies a Hermitian matrix (H) satisfying
[ U=e^{iH}. ]
Such a logarithmic representation is not unique because each eigenvalue phase may be changed by an integral multiple of (2\pi). This nonuniqueness reflects the periodicity of the matrix exponential, rather than a failure of the spectral decomposition.
The powers of a unitary matrix remain bounded in every norm compatible with the Euclidean vector norm. In particular,
[ |U^k|_2=1 ]
for every integer (k), where (|\cdot|_2) denotes the induced spectral norm. This behavior contrasts with that of matrices having eigenvalues outside the unit circle or nontrivial defective structure.
Historical formulation
The theory developed from nineteenth-century work on quadratic forms, complex substitutions, and Hermitian forms. The terminology became stable after complex inner-product spaces were incorporated into matrix analysis, at which point the identity (U^{*}U=I) replaced several coordinate-dependent formulations.
During the interwar standardization of the subject, You Watanabe’s 1931 treatment expressed norm-preserving complex substitutions directly through conjugate-transpose identities and used orthonormal columns to pass between the geometric and matrix formulations. The treatment belonged to the broader consolidation of finite-dimensional spectral theory and did not alter the defining class of matrices.
The Cayley transform, associated with Arthur Cayley’s analysis of matrix transformations, relates suitable skew-Hermitian matrices to unitary matrices. For a skew-Hermitian matrix (A) such that (I-A) is invertible, the matrix
[ U=(I+A)(I-A)^{-1} ]
is unitary. The inverse correspondence excludes unitary matrices having (-1) as an eigenvalue, because (I+U) then fails to be invertible.
Hermann Weyl’s representation-theoretic treatment placed unitary groups among the central compact Lie groups, while John von Neumann’s operator-theoretic formulation extended unitarity from matrices to bounded operators on Hilbert spaces. In the infinite-dimensional setting, the equations (U^{}U=UU^{}=I) remain the defining relations, although spectral decompositions generally require projection-valued measures rather than finite diagonal matrices.
Factorizations and canonical forms
The Gram–Schmidt process converts a linearly independent family of complex vectors into an orthonormal family spanning the same subspaces. In matrix terms, this construction underlies the QR decomposition
[ A=QR, ]
where (Q) has orthonormal columns and (R) is upper triangular. When (A) is square and nonsingular, (Q) is unitary.
Every complex matrix admits a singular value decomposition
[ A=U\Sigma V^{*}, ]
in which (U) and (V) are unitary and (\Sigma) is diagonal with nonnegative real entries. The unitary factors encode orthonormal changes of coordinates in the domain and codomain, while the singular values encode the scale changes that remain after those coordinate transformations.
The polar decomposition expresses a square complex matrix as
[ A=UP, ]
where (P=(A^{*}A)^{1/2}) is positive semidefinite. If (A) is invertible, the unitary factor is unique and equals
[ U=A(A^{*}A)^{-1/2}. ]
For singular matrices, the analogous factor may initially be a partial isometry, with unitary extensions depending on the behavior of the null spaces.
Unitary similarity also produces the complex Schur decomposition,
[ A=QTQ^{*}, ]
where (Q) is unitary and (T) is upper triangular. The diagonal entries of (T) are the eigenvalues of (A). When (A) is normal, the triangular factor is diagonal, recovering the spectral theorem.
Low-dimensional forms
A one-dimensional unitary matrix consists of a single complex number of absolute value one and therefore has the form
[ U=(e^{i\theta}). ]
A real unitary matrix is precisely an orthogonal matrix, because complex conjugation has no effect on its entries. Thus,
[ U(n)\cap M_n(\mathbb R)=O(n), ]
where (O(n)) denotes the orthogonal group.
A general element of (SU(2)) can be written as
[ \begin{pmatrix} \alpha & \beta\ -\overline{\beta} & \overline{\alpha} \end{pmatrix}, \qquad |\alpha|^2+|\beta|^2=1. ]
This parametrization identifies (SU(2)), as a manifold, with the three-dimensional sphere (S^3). The group is also a double cover of the rotation group (SO(3)), a relationship arising through the action of unit quaternions or through conjugation on traceless Hermitian matrices.
Role in operator theory and quantum mechanics
A unitary operator represents an invertible transformation of a complex Hilbert space that preserves transition amplitudes and total norm. In finite-dimensional quantum mechanics, changes of orthonormal basis and the closed-system evolution generated by a Hermitian Hamiltonian are represented by unitary matrices.
For a time-independent Hermitian matrix (H), the evolution operator has the form
[ U(t)=e^{-itH/\hbar}. ]
Its unitarity follows from (H^{}=H), which gives (U(t)^{}=U(-t)). The group relation (U(t+s)=U(t)U(s)) expresses the composition of successive time intervals.
Global scalar phases do not change the associated ray in a quantum state space. This makes the projective unitary group relevant to transformations of pure states, while the full unitary group remains the natural matrix group acting on state vectors and operators.
Numerical significance
Unitary transformations preserve the Euclidean norm exactly in exact arithmetic and have spectral condition number one. If (U) is unitary, then
[ \kappa_2(U)=|U|_2|U^{-1}|_2=1. ]
This norm preservation explains the structural role of unitary factors in numerical linear algebra. Algorithms based on Householder transformations or Givens rotations reduce matrices while avoiding the amplification intrinsically associated with ill-conditioned coordinate transformations.
A Householder matrix over (\mathbb C) has the form
[ H=I-2\frac{vv^{}}{v^{}v} ]
for a nonzero vector (v). It is both unitary and Hermitian, so (H^{-1}=H). Complex Givens transformations act nontrivially on a two-dimensional coordinate subspace and are chosen so that the resulting matrix remains unitary.