skew-Hermitian matrix

Definition. A square matrix $A$ with complex entries is skew-Hermitian, if

$$A^{*}=-A.$$

Here $A^{\ast }=\overline{A^{\text{T}}}$, $A^{\text{T}}$ is the transpose of $A$, and $\overline{A}$ is is the complex conjugate of the matrix $A$.

Properties.

1. The trace of a skew-Hermitian matrix is http://planetmath.org/node/2017imaginary.

2. The eigenvalues of a skew-Hermitian matrix are http://planetmath.org/node/2017imaginary.

Proof. Property (1) follows directly from property (2) since the trace is the sum of the eigenvalues. But one can also give a simple proof as follows. Let $x_{ij}$ and $y_{ij}$ be the real respectively imaginary parts of the elements in $A$. Then the diagonal elements of $A$ are of the form $x_{kk}+i{y}_{kk}$, and the diagonal elements in $A^{\ast }$ are of the form $-x_{kk}+i{y}_{kk}$. Hence $x_{kk}$, i.e., the real part for the diagonal elements in $A$ must vanish, and property (1) follows. For property (2), suppose $A$ is a skew-Hermitian matrix, and $x$ an eigenvector corresponding to the eigenvalue $\lambda$, i.e.,

$Ax$ $=$ $\lambda x.$ (1)

Here, $x$ is a complex column vector. Since $x$ is an eigenvector, $x$ is not the zero vector, and $x^{\ast }x>0$. Without loss of generality we can assume $x^{\ast }x=1$. Thus

$\overline{\lambda }$ $=$ $x^{\ast }\overline{\lambda }x$

$=$ ${(x^{\ast }\lambda x)}^{\ast }$

$=$ ${(x^{\ast }Ax)}^{\ast }$

$=$ $x^{\ast }{A}^{\ast }x$

$=$ $x^{\ast }(-A)x$

$=$ $-x^{\ast }\lambda x$

$=$ $-\lambda .$

Hence the eigenvalue $\lambda$ corresponding to $x$ is http://planetmath.org/node/2017imaginary. $\mathrm{\square }$

Title	skew-Hermitian matrix
Canonical name	SkewHermitianMatrix
Date of creation	2013-03-22 13:36:14
Last modified on	2013-03-22 13:36:14
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	21
Author	matte (1858)
Entry type	Definition
Classification	msc 15A57
Synonym	anti-Hermitian matrix
Related topic	HermitianMatrix
Related topic	SymmetricMatrix
Related topic	SkewSymmetricMatrix