skewHermitian matrix
Definition. A square matrix^{} $A$ with complex entries is skewHermitian, 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 skewHermitian matrix is http://planetmath.org/node/2017imaginary.

2.
The eigenvalues^{} of a skewHermitian 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 skewHermitian 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  skewHermitian matrix 

Canonical name  SkewHermitianMatrix 
Date of creation  20130322 13:36:14 
Last modified on  20130322 13:36:14 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  21 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 15A57 
Synonym  antiHermitian matrix 
Related topic  HermitianMatrix 
Related topic  SymmetricMatrix 
Related topic  SkewSymmetricMatrix 