skew-Hermitian matrix

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

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

Here $A^{\ast }=\overline{A^{\text{T}}}$, $A^{\text{T}}$ is the transpose of $A$, and $\overline{A}$ isis 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 arehttp://planetmath.org/node/2017imaginary.

Proof. Property (1) follows directly from property (2) since thetrace is the sum of the eigenvalues. But one can also give a simple proofas follows. Let $x_{ij}$ and $y_{ij}$ be thereal respectively imaginary parts of the elements in $A$.Then the diagonal elements of $A$ are of theform $x_{kk}+i{y}_{kk}$, and the diagonal elements in $A^{\ast }$are of the form $-x_{kk}+i{y}_{kk}$. Hence $x_{kk}$, i.e., the realpart for the diagonal elements in $A$ must vanish, andproperty (1) follows.For property (2), suppose$A$ is a skew-Hermitian matrix, and $x$ aneigenvector 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 $\lambda$ correspondingto $x$ is http://planetmath.org/node/2017imaginary. $\mathrm{\square }$