# skew-Hermitian matrix

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

 $A^{*}=-A.$

Here $A^{\ast}=\overline{A\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt}$, $A\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt$ is the transpose of $A$, and $\overline{A}$ is is the complex conjugate of the matrix $A$.

## Properties.

1. 1.

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

2. 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 $x_{kk}+iy_{kk}$, 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 http://planetmath.org/node/2017imaginary. $\Box$

Title skew-Hermitian matrix SkewHermitianMatrix 2013-03-22 13:36:14 2013-03-22 13:36:14 matte (1858) matte (1858) 21 matte (1858) Definition msc 15A57 anti-Hermitian matrix HermitianMatrix SymmetricMatrix SkewSymmetricMatrix