unitaries

Unitary matrices are complex square matrices U that satisfy U* U = I, where U* denotes the conjugate transpose. Equivalently, U is invertible and U^{-1} = U*. As linear operators on a complex inner product space, unitary matrices preserve the standard inner product, so for all vectors x and y, ⟨Ux, Uy⟩ = ⟨x, y⟩.

Because they preserve inner products, unitary transformations also preserve norms and angles. The eigenvalues of a unitary matrix have absolute value 1, and the determinant of a unitary matrix has absolute value 1. The rows and columns of a unitary matrix form orthonormal bases of the complex vector space.

All unitary matrices of size n form the unitary group U(n) under multiplication. The subgroup consisting of unitary matrices with determinant 1 is the special unitary group SU(n). These groups are compact Lie groups with dimension n^2 in the real sense.

Examples of unitary matrices include the identity matrix, diagonal matrices with entries on the unit circle, and permutation matrices. The Fourier matrix F with entries F_{jk} = ω^{jk}/√n, where ω = e^{2πi/n}, is unitary and widely used in signal processing and numerical analysis.

Unitary matrices play a central role in quantum mechanics and quantum information, where they describe the evolution of quantum states and quantum gates. They are also fundamental in signal processing, numerical linear algebra, and representation theory. In infinite dimensions, unitary operators on a Hilbert space generalize these concepts and underpin much of functional analysis and quantum theory.