Schiefhermitean matrix

A skew-Hermitian matrix or anti-Hermitian matrix is a mathematical object from linear algebra . This special type of square matrices with complex coefficients is converted into their adjoint matrix with respect to the complex standard scalar product when the coefficients are reflected on the main diagonal . These matrices are named after the mathematician Charles Hermite .

definition

A square matrix is called skewed Hermitian if it is equal to its negative adjoint, that means ${\displaystyle B\in \mathbb {C} ^{n\times n}}$

${\displaystyle B=-B^{H}=-{\overline {B}}^{T}}$.

The following applies to the entries of a lopsided Hermitian matrix

${\displaystyle b_{jk}=-{\overline {b_{kj}}}}$.

Examples

• The matrix

${\displaystyle {\begin{pmatrix}3i&2+i\\-2+i&i\end{pmatrix}}}$

with as the imaginary unit is askew Hermitian. ${\displaystyle i^{2}=-1}$

• The matrices ${\displaystyle 2\times 2}$

${\displaystyle {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}\mapsto \mathrm {i} ,\quad {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\mapsto \mathrm {j} ,\quad {\begin{pmatrix}0&i\\i&0\end{pmatrix}}\mapsto \mathrm {k} ,}$

which can be mapped to the quaternionic generators as shown are skewed Hermitian and free of traces .

properties

• The main diagonal elements are purely imaginary.
• The real part is skew symmetrical , the imaginary part is symmetrical .
• If Hermitian is wrong, then Hermitian is . ${\displaystyle B}$ ${\displaystyle iB}$
• The eigenvalues ​​of skewed Hermitian matrices are purely imaginary , the eigenvectors form an orthonormal system for the standard Hermitian form .
• Schiefhermitean matrices can always be diagonalized .
• In the real world , the terms skewed Hermitian and skew symmetrical coincide. Real skew-symmetrical matrices can be brought into a block-diagonal form by real base change with -blocks ${\displaystyle (2\times 2)}$

${\displaystyle {\begin{pmatrix}0&r\\-r&0\end{pmatrix}}\ ,\ r\in \mathbb {R} }$.

• If Hermitian is wrong, then Hermitian is even and wrong Hermitian is odd . ${\displaystyle B}$ ${\displaystyle B^{k}}$ ${\displaystyle k}$ ${\displaystyle k}$
• If Hermitian is wrong, then it is unitary . ${\displaystyle B}$ ${\displaystyle e^{B}}$
• Any square matrix can be uniquely written as the sum of a Hermitian matrix and a skewed Hermitian matrix : ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle C=A+B}$

with and . ${\displaystyle A=(C+C^{H})/2}$ ${\displaystyle B=(C-C^{H})/2}$

The Lie algebra of the skew Hermitian matrices

The commutator of skewed Hermitian matrices is skewed Hermitian again. The lopsided Hermitian matrices thus form a Lie algebra , which is denoted by. ${\displaystyle (n\times n)}$ ${\displaystyle {\mathfrak {u}}(n)}$

${\displaystyle {\mathfrak {u}}(n)=\left\{X\in \mathrm {Mat} (n,\mathbb {C} )\colon X+{\overline {X}}^{\mathrm {T} }=0\right\}}$

is the Lie algebra of the Lie group of unitary matrices

${\displaystyle U(n)=\left\{A\in \mathrm {GL} (n,\mathbb {C} )\colon A{\overline {A}}^{\mathrm {T} }=E_{n}\right\}}$.

literature

• Hans-Joachim Kowalsky , Gerhard O. Michler : Lineare Algebra. de Gruyter, Berlin et al. 2003, ISBN 3-11-017963-6 .

Individual evidence

1. ↑ Hans-Joachim Kowalsky, Gerhard O. Michler: Lineare Algebra. de Gruyter, 2003, p. 182.