Skew-Hermitian matrix

In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) "A" is said to be skew-Hermitian or antihermitian if its conjugate transpose "A"^* is also its negative. That is, if it satisfies the relation::"A"^* = −"A"or in component form, if "A" = ("a"_"i,j")::$a\_\{i,j\}\; =\; -overline\{a\_\{j,i$for all "i" and "j".

Examples

For example, the following matrix is skew-Hermitian::

Properties

* The eigenvalues of a skew-Hermitian matrix are all purely imaginary. Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
* All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, ie. on the imaginary axis. This definition includes the number "0i".
* If "A" is skew-Hermitian, then "iA" is Hermitian
* If "A, B" are skew-Hermitian, then "aA + bB" is skew-Hermitian for all real scalars "a, b".
* If "A" is skew-Hermitian, then "A"^"2k" is Hermitian for all positive integers "k".
* If "A" is skew-Hermitian, then "A" raised to an odd power is skew-Hermitian.
* If "A" is skew-Hermitian, then e^"A" is unitary.
* The difference of a matrix and its conjugate transpose ($C\; -\; C^*$) is skew-Hermitian.
* An arbitrary (square) matrix "C" can be written as the sum of a Hermitian matrix "A" and a skew-Hermitian matrix "B":::$C\; =\; A+B\; quadmbox\{with\}quad\; A\; =\; frac\{1\}\{2\}(C\; +\; C^*)\; quadmbox\{and\}quad\; B\; =\; frac\{1\}\{2\}(C\; -\; C^*).$
* The space of skew-Hermitian matrices forms the Lie algebra u("n") of the Lie group U("n").

ee also

*skew-symmetric matrix
*Hermitian matrix
*normal matrix
*unitary matrix