Conjugatesymmetric

Conjugate symmetric is a term used in linear algebra to describe complex matrices that are equal to their own conjugate transpose. Formally, a complex n-by-n matrix A is conjugate symmetric if A = A*, where A* denotes the conjugate transpose (also called the Hermitian transpose). Equivalently, for all i and j, A_ij = conj(A_ji). This condition is often referred to as being Hermitian in many texts.

Properties: Conjugate-symmetric (Hermitian) matrices have several important features. All eigenvalues of A are real, and A is diagonalizable by a unitary matrix: there exists a unitary U such that A = U Λ U^H, where Λ is diagonal with the real eigenvalues. The trace is real, and the determinant is real. For any vector x, the scalar x^H A x is real and nonnegative if A is positive semidefinite.

Examples: A = [[1, i], [-i, 2]] is conjugate symmetric since its conjugate transpose equals itself. Another typical form is A = [[a, z], [conj(z), b]] with a and b real and z any complex number.

Relation to other notions: If a matrix is complex symmetric (A^T = A) but not Hermitian, its eigenvalues need not be real. The term conjugate symmetric is closely related to Hermitian operators in functional analysis and physics, where observables are modeled by Hermitian operators.

Applications: Conjugate-symmetric matrices appear as Hermitian operators in quantum mechanics, as well as in signal processing and statistics in the form of covariance or correlation matrices, which are Hermitian. They provide stable spectral decompositions and real-valued quadratic forms.