unitary

Unitary is a term used in linear algebra, functional analysis, and physics to describe a property of linear operators and matrices that preserves inner products and related structures.

In linear algebra, a unitary matrix U is a complex square matrix that satisfies U* U = U U* = I, where U* is the conjugate transpose. Equivalently, U is invertible and U^{-1} = U*. This implies that unitary matrices preserve the complex inner product, and hence norms and angles between vectors. Over the real numbers, orthogonal matrices satisfy U^T U = I and are the real counterparts of unitary matrices. The eigenvalues of a unitary matrix have absolute value 1, and det(U) has magnitude 1. The set of all n-by-n unitary matrices forms the unitary group U(n).

In infinite dimensions, a unitary operator U on a complex Hilbert space H is a linear bijection that preserves the inner product: U* U = U U* = I. The inverse of a unitary operator is its adjoint, U^{-1} = U*. The collection of all unitary operators on H is denoted U(H). For finite-dimensional spaces, U(H) is isomorphic to U(n).

Unitary transformations are fundamental in physics, especially quantum mechanics, where they describe time evolution and quantum gates. Unitary evolution preserves total probability, with time evolution given by e^{-iHt/ħ} for a Hamiltonian H. In signal processing, unitary transforms, such as the Fourier transform (up to normalization), preserve signal energy.

In algebra, an element u of a ring with involution is called unitary if u* u = u u* = 1. The term derives from Latin unitas, meaning unity or wholeness.