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Homeskew-Hermitian matrix

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# skew-Hermitian matrix

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

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

Here $A^{\ast}=\overline{A^{{\mbox{\scriptsize{T}}}}}$, $A^{{\mbox{\scriptsize{T}}}}$ 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 imaginary.

2. 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 $A$.
Then the diagonal elements of $A$ are of the
form $x_{{kk}}+iy_{{kk}}$, 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 $A$ 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.,

$\displaystyle Ax$ | $\displaystyle=$ | $\displaystyle\lambda x.$ | (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

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

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

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

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

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

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

$\displaystyle=$ | $\displaystyle-\lambda.$ |

Hence the eigenvalue $\lambda$ corresponding to $x$ is imaginary. $\Box$

## Mathematics Subject Classification

15A57*no label found*

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