Hermitianinen

Hermitianinen is a term used in linear algebra to describe matrices or linear operators that are equal to their own conjugate transpose. In English mathematics, the standard term is Hermitian. For a complex square matrix A, the conjugate transpose is denoted A†, or A*, and A is Hermitian if A = A†.

This equality implies that all diagonal entries of a Hermitian matrix are real, and the off-diagonal entries satisfy aij = conj(aji) for all i and j. In other words, the matrix is complex-conjugate symmetric with respect to its diagonal.

A central property of Hermitian matrices is that their eigenvalues are real. Hermitian matrices are diagonalizable by a unitary matrix: there exists a unitary U such that U†AU = diag(λ1, ..., λn) with real λi. Eigenvectors corresponding to distinct eigenvalues are orthogonal, and one can choose an orthonormal basis of eigenvectors.

Real symmetric matrices are a subset of Hermitian matrices, since a real matrix is Hermitian precisely when it is symmetric (A^T = A). In the broader, infinite-dimensional setting of functional analysis, a Hermitian (often called self-adjoint) operator on a complex inner product space satisfies ⟨Ax, y⟩ = ⟨x, Ay⟩ for all vectors x and y in its domain. The spectral theorem extends this structure to provide a real-spectrum representation.

Applications of Hermitian objects are widespread, including quantum mechanics (observables are represented by Hermitian operators), signal processing, and numerical linear algebra. In these contexts, the real eigenvalues and orthogonal eigenvectors underpin stable, interpretable spectral decompositions.