PlanetMath: skew-Hermitian matrix

(more info)

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skew-Hermitian matrix

(Definition)

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

$\displaystyle A^* = -A.$

Here $A^\ast=\overline{A\hspace{0.04cm} ^{\mbox{\scriptsize {T}}} \hspace{0.02cm}}$ , $A\hspace{0.04cm} ^{\mbox{\scriptsize {T}}} \hspace{0.02cm}$ is the transpose of

, and $\overline{A}$ is is the complex conjugate of the matrix

Properties.

The trace of a skew-Hermitian matrix is imaginary.
The eigenvalues of a skew-Hermitian matrix are imaginary.

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 . Then the diagonal elements of are of the form $x_{kk} + i y_{kk}$ , and the diagonal elements in $A^\ast$ are of the form $-x_{kk} + iy_{kk}$ . Hence $x_{kk}$ , i.e., the real part for the diagonal elements in must vanish, and property (1) follows. For property (2), suppose is a skew-Hermitian matrix, and an eigenvector corresponding to the eigenvalue $\lambda$ , i.e.,

$\displaystyle Ax$

$\displaystyle =$

$\displaystyle \lambda x.$

(1)

Here,

is a complex column vector. Since

is an eigenvector,

is not the zero vector, and $x^\ast x > 0$ . Without loss of generality we can assume $x^\ast x =1$ . Thus

$\displaystyle \overline{\lambda}$	$\displaystyle =$	$\displaystyle x^\ast \overline{\lambda} x$
	$\displaystyle =$	$\displaystyle ( x^\ast \lambda x )^\ast$
	$\displaystyle =$	$\displaystyle (x^\ast A x )^\ast$
	$\displaystyle =$	$\displaystyle x^\ast A^\ast x$
	$\displaystyle =$	$\displaystyle x^\ast (-A) x$
	$\displaystyle =$	$\displaystyle -x^\ast \lambda x$
	$\displaystyle =$	$\displaystyle - \lambda .$