Skew-Hermitian matrix

Template:Short description

In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix.<ref>Template:Harvtxt, §4.1.1; Template:Harvtxt, §3.2</ref> That is, the matrix <math>A</math> is skew-Hermitian if it satisfies the relation

Template:Equation box 1

where <math>A^\textsf{H}</math> denotes the conjugate transpose of the matrix <math>A</math>. In component form, this means that

Template:Equation box 1</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}

for all indices <math>i</math> and <math>j</math>, where <math>a_{ij}</math> is the element in the <math>i</math>-th row and <math>j</math>-th column of <math>A</math>, and the overline denotes complex conjugation.

Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers.<ref name=HJ85S412>Template:Harvtxt, §4.1.2</ref> The set of all skew-Hermitian <math>n \times n</math> matrices forms the <math>u(n)</math> Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm.

Note that the adjoint of an operator depends on the scalar product considered on the <math>n</math> dimensional complex or real space <math>K^n</math>. If <math>(\cdot\mid\cdot) </math> denotes the scalar product on <math> K^n</math>, then saying <math> A</math> is skew-adjoint means that for all <math>\mathbf u, \mathbf v \in K^n</math> one has <math> (A \mathbf u \mid \mathbf v) = - (\mathbf u \mid A \mathbf v)</math>.

Imaginary numbers can be thought of as skew-adjoint (since they are like <math>1 \times 1</math> matrices), whereas real numbers correspond to self-adjoint operators.

ExampleEdit

For example, the following matrix is skew-Hermitian <math display="block"> A = \begin{bmatrix} -i & +2 + i \\ -2 + i & 0 \end{bmatrix}</math> because <math display="block">

 -A =
 \begin{bmatrix} i & -2 - i \\ 2 - i & 0 \end{bmatrix} =
 \begin{bmatrix}
   \overline{-i}    & \overline{-2 + i} \\
   \overline{2 + i} & \overline{0}
 \end{bmatrix} =
 \begin{bmatrix}
   \overline{-i}     & \overline{2 + i} \\
   \overline{-2 + i} &     \overline{0}
 \end{bmatrix}^\mathsf{T} =
 A^\mathsf{H}

</math>

PropertiesEdit

The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.<ref>Template:Harvtxt, §2.5.2, §2.5.4</ref>
All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary; i.e., on the imaginary axis (the number zero is also considered purely imaginary).<ref>Template:Harvtxt, Exercise 3.2.5</ref>
If <math>A</math> and <math>B</math> are skew-Hermitian, then Template:Tmath is skew-Hermitian for all real scalars <math>a</math> and <math>b</math>.<ref name=HJ85S411>Template:Harvtxt, §4.1.1</ref>
<math>A</math> is skew-Hermitian if and only if <math>i A</math> (or equivalently, <math>-i A</math>) is Hermitian.<ref name=HJ85S411/>
<math>A</math> is skew-Hermitian if and only if the real part <math>\Re{(A)}</math> is skew-symmetric and the imaginary part <math>\Im{(A)}</math> is symmetric.
If <math>A</math> is skew-Hermitian, then <math>A^k</math> is Hermitian if <math>k</math> is an even integer and skew-Hermitian if <math>k</math> is an odd integer.
<math>A</math> is skew-Hermitian if and only if <math>\mathbf{x}^\mathsf{H} A \mathbf{y} = -\overline{\mathbf{y}^\mathsf{H} A \mathbf{x}}</math> for all vectors <math>\mathbf x, \mathbf y</math>.
If <math>A</math> is skew-Hermitian, then the matrix exponential <math>e^A</math> is unitary.
The space of skew-Hermitian matrices forms the Lie algebra <math>u(n)</math> of the Lie group <math>U(n)</math>.

Decomposition into Hermitian and skew-HermitianEdit

The sum of a square matrix and its conjugate transpose <math>\left(A + A^\mathsf{H}\right)</math> is Hermitian.
The difference of a square matrix and its conjugate transpose <math>\left(A - A^\mathsf{H}\right)</math> is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
An arbitrary square matrix <math>C</math> can be written as the sum of a Hermitian matrix <math>A</math> and a skew-Hermitian matrix <math>B</math>: <math display="block">C = A + B \quad\mbox{with}\quad A = \frac{1}{2}\left(C + C^\mathsf{H}\right) \quad\mbox{and}\quad B = \frac{1}{2}\left(C - C^\mathsf{H}\right)</math>

NotesEdit

ReferencesEdit