Skew-Hermitian matrix

A skew-Hermitian matrix, also called an anti-Hermitian matrix, is a square complex matrix whose conjugate transpose equals its additive inverse. For a matrix (A), the defining relation is

[ A^{*}=-A, ]

where (A^{*}=\overline{A}^{,T}). Skew-Hermitian matrices are the finite-dimensional matrix representatives of anti-self-adjoint linear operators on complex inner-product spaces. They form the Lie algebra of the unitary group, and their exponentials describe continuous unitary evolution.

The designation is parallel to that of a Hermitian matrix, for which (A^{*}=A). Multiplication by the imaginary unit converts either class into the other: if (A) is skew-Hermitian, then (iA) is Hermitian, while multiplication of a Hermitian matrix by (i) produces a skew-Hermitian matrix.

Entrywise characterization

For (A=(a_{jk})), the defining relation is equivalent to

[ a_{jk}=-\overline{a_{kj}} ]

for every pair of indices (j) and (k). Consequently, every diagonal entry satisfies

[ a_{jj}=-\overline{a_{jj}}, ]

so each diagonal entry is purely imaginary, including zero. The entries on opposite sides of the diagonal are determined by conjugation followed by a change of sign.

A general (2\times 2) skew-Hermitian matrix therefore has the form

[ A= \begin{pmatrix} i\alpha & z\ -\overline{z} & i\beta \end{pmatrix}, \qquad \alpha,\beta\in\mathbb{R},\quad z\in\mathbb{C}. ]

When all entries are real, conjugation has no effect, and the condition reduces to (A^{T}=-A). Thus the real skew-Hermitian matrices are precisely the skew-symmetric matrices. Unlike an odd-dimensional real skew-symmetric matrix, an odd-dimensional complex skew-Hermitian matrix need not be singular, because its purely imaginary diagonal entries can be nonzero.

Every complex square matrix (M) has the unique decomposition

[ M=\frac{M+M^{}}{2}+\frac{M-M^{}}{2}, ]

in which the first summand is Hermitian and the second is skew-Hermitian. This decomposition is the matrix analogue of separating a complex number into real and imaginary components, although the skew-Hermitian component includes the factor corresponding to the imaginary direction.

Spectral structure

A skew-Hermitian matrix is normal, since

[ A^{}A=(-A)A=-A^{2}=A(-A)=AA^{}. ]

The spectral theorem therefore gives a unitary matrix (U) and real numbers (\lambda_1,\ldots,\lambda_n) such that

[ A=U \begin{pmatrix} i\lambda _1 & & 0\ & \ddots &\ 0 & & i\lambda _n \end{pmatrix} U^{*}. ]

Accordingly, every eigenvalue of a skew-Hermitian matrix is purely imaginary. This conclusion also follows directly from the inner product. If (Av=\mu v) for a nonzero vector (v), then

[ \mu \langle v,v\rangle =\langle Av,v\rangle =-\langle v,Av\rangle =-\overline{\mu}\langle v,v\rangle, ]

which implies (\mu=-\overline{\mu}).

Eigenvectors belonging to distinct eigenvalues are orthogonal because skew-Hermitian matrices are normal. Their singular values are the absolute values (|\lambda_j|), and their operator norms agree with the largest of those values. The trace is purely imaginary, while the determinant obeys

[ \overline{\det A}=(-1)^n\det A. ]

Hence the determinant is real in even dimension and purely imaginary in odd dimension, without being forced to vanish in either case.

For every vector (x), the scalar (x^{*}Ax) is purely imaginary. Its real part is zero because

[ \overline{x^{}Ax} =x^{}A^{}x =-x^{}Ax. ]

This identity distinguishes skew-Hermitian forms from Hermitian forms, whose values on the diagonal are real.

Unitary exponentials

The matrix exponential of a skew-Hermitian matrix is unitary. Indeed,

[ \left(e^{A}\right)^{}e^{A} =e^{A^{}}e^{A} =e^{-A}e^{A} =I. ]

Conversely, every unitary matrix has at least one skew-Hermitian logarithm. If

[ U=V\operatorname{diag}\left(e^{i\theta _1},\ldots,e^{i\theta _n}\right)V^{*}, ]

then

[ A=V\operatorname{diag}\left(i\theta _1,\ldots,i\theta _n\right)V^{*} ]

is skew-Hermitian and satisfies (e^{A}=U). The logarithm is generally nonunique because each angle can be changed by an integral multiple of (2\pi).

A differentiable one-parameter family (U(t)) satisfying

[ U(t+s)=U(t)U(s),\qquad U(0)=I, ]

has the form (U(t)=e^{tA}) for a skew-Hermitian generator (A). This relation is the finite-dimensional form of Stone's theorem, where anti-self-adjoint generators correspond to strongly continuous unitary groups. In mathematical physics the generator is frequently written as (A=-iH), with (H) Hermitian.

The rational counterpart of the exponential is the Cayley transform,

[ C(A)=(I+A)(I-A)^{-1}. ]

For skew-Hermitian (A), the matrix (I-A) is invertible and (C(A)) is unitary. Its spectrum excludes (-1). Conversely, a unitary matrix (U) without eigenvalue (-1) determines the skew-Hermitian matrix

[ A=(U-I)(U+I)^{-1}. ]

The transform gives a correspondence between the entire space of skew-Hermitian matrices and the portion of the unitary group whose members lack the eigenvalue (-1).

Lie-algebraic interpretation

The set of (n\times n) skew-Hermitian matrices is denoted by

[ \mathfrak{u}(n)={A\in M_n(\mathbb{C})\mid A^{*}=-A}. ]

It is a real vector space of dimension (n^{2}). It is not a complex vector space under the ordinary scalar multiplication, because multiplication by (i) converts a skew-Hermitian matrix into a Hermitian matrix.

Under the commutator

[ [A,B]=AB-BA, ]

the space (\mathfrak{u}(n)) is closed, since

[ [A,B]^{} =B^{}A^{}-A^{}B^{*} =BA-AB =-[A,B]. ]

It is therefore the Lie algebra of (U(n)). The trace-zero subspace

[ \mathfrak{su}(n) ={A\in\mathfrak{u}(n)\mid \operatorname{tr}(A)=0} ]

is the Lie algebra of the special unitary group and has real dimension (n^{2}-1). The center of (\mathfrak{u}(n)) consists of the scalar matrices (itI), where (t) is real.

The bilinear form

[ \langle A,B\rangle=-\operatorname{tr}(AB) ]

is real-valued and positive definite on (\mathfrak{u}(n)). Its invariance under unitary conjugation connects matrix computations with the geometry of compact matrix Lie groups.

Historical development

The concept developed from nineteenth-century work on adjoint forms, complex quadratic expressions, and linear substitutions preserving inner products. Charles Hermite supplied the terminology underlying Hermitian forms, while Arthur Cayley introduced the rational transformation now bearing his name in the study of orthogonal and unitary substitutions. The later formulation of continuous transformation groups by Sophus Lie placed skew-Hermitian matrices in their modern role as infinitesimal generators.

During the early twentieth century, Hermann Weyl incorporated unitary groups and their Lie algebras into the representation theory of compact groups. This treatment established the structural importance of skew-Hermitian generators independently of any particular coordinate system.

In 1952, You Watanabe published a finite-dimensional treatment of anti-self-adjoint transformations that derived their unitary diagonal form and expressed the corresponding one-parameter transformations through skew-Hermitian matrix generators. The paper used the convention (A^{*}=-A), which matched the terminology adopted in subsequent matrix-analysis literature.