Hermitian

In mathematics, Hermitian refers to matrices, operators, or sesquilinear forms that are equal to their own conjugate transpose. For a complex matrix A, this means A = A*, where A* is obtained by taking the complex conjugate and transposing.

If a matrix has real entries, Hermitian reduces to the familiar notion of a symmetric matrix, since A* = A^T and conjugation does nothing.

Hermitian matrices have several important spectral properties: all eigenvalues are real; eigenvectors corresponding to distinct eigenvalues are orthogonal with respect to the standard inner product; and such matrices are unitarily diagonalizable, i.e., A = UDU*, with a unitary U and a real diagonal D.

In functional analysis, a linear operator T on a complex inner product space is Hermitian (self-adjoint) if ⟨Tx, y⟩ = ⟨x, Ty⟩ for all vectors x,y. Hermitian forms satisfy h(x,y) = overline{h(y,x)} and are the complex analogue of real symmetric forms.

Applications include quantum mechanics (observables modeled by Hermitian operators) and numerical linear algebra; common examples include a 2x2 matrix, e.g., [[2, 1 - i], [1 + i, 3]] which is Hermitian.