SkewHermitian

Skew-Hermitian, sometimes called anti-Hermitian, refers to a property of complex square matrices A for which the conjugate transpose equals its negative: A* = -A. Here A* denotes the transpose followed by complex conjugation.

Basic consequences of the definition include constraints on the entries and the spectrum. The diagonal entries of a skew-Hermitian matrix must be purely imaginary. For the off-diagonal entries, if aij is an element, then aji = -conj(aij). The trace of a skew-Hermitian matrix is purely imaginary. The eigenvalues of a skew-Hermitian matrix are all purely imaginary (or zero). Skew-Hermitian matrices are normal, meaning A*A = AA*, and thus they are diagonalizable by a unitary matrix. In particular, there exists a unitary U such that A = UDU*, where D is diagonal with eigenvalues of the form iλj, with λj real.

A useful relation is that iA is Hermitian, since (iA)* = -iA* = iA. This places skew-Hermitian matrices as the real Lie algebra u(n) of the unitary group U(n); the Lie bracket is given by [A,B] = AB − BA. The matrix exponential of a skew-Hermitian matrix is unitary: eA is unitary because (eA)† = eA* = e−A and eA e−A = I.

A simple example in 2x2 form is A = [ [0, −1], [1, 0] ], which satisfies A* = −A and has eigenvalues ±i. Skew-Hermitian matrices arise naturally in areas such as quantum mechanics and differential geometry, where unitary evolution and unitary representations play central roles.