skew-Hermitian matrix
Definition. A square matrix with complex entries is
skew-Hermitian, if
Here , is the transpose of , and is
is the complex conjugate
of the matrix .
Properties.
-
1.
The trace of a skew-Hermitian matrix is http://planetmath.org/node/2017imaginary.
-
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 and be the
real respectively imaginary parts of the elements in .
Then the diagonal elements of are of the
form , and the diagonal elements in
are of the form . Hence , 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 , i.e.,
(1) |
Here, is a complex column vector.
Since is an eigenvector, is not the zero vector
, and
. Without loss of generality we can assume .
Thus
Hence the eigenvalue corresponding to is http://planetmath.org/node/2017imaginary.
Title | skew-Hermitian matrix |
---|---|
Canonical name | SkewHermitianMatrix |
Date of creation | 2013-03-22 13:36:14 |
Last modified on | 2013-03-22 13:36:14 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 21 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 15A57 |
Synonym | anti-Hermitian matrix |
Related topic | HermitianMatrix |
Related topic | SymmetricMatrix |
Related topic | SkewSymmetricMatrix |