Skew-Hermitian matrix

In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is the negative of the original matrix.^[1] That is, the matrix A is skew-Hermitian if it satisfies the relation

A^{\dagger }=-A,\;

where $\dagger$ denotes the conjugate transpose of a matrix. In component form, this means that

a_{{i,j}}=-\overline {a_{{j,i}}},

for all i and j, where a_i,j is the i,j-th entry of A, and the overline denotes complex conjugation.

Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers.^[2] The set of all skew-Hermitian n×n matrices forms the u(n) Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm.

Note that the adjoint of an operator depends on the scalar product considered on the $n$ dimensional complex or real space $K^{n}$ . If $(\cdot |\cdot )$ denotes the scalar product on $K^{n}$ , then saying $A$ is skew-adjoint means that for all $u,v\in K^{n}$ one has $(Au|v)=-(u|Av)\,.$

Imaginary numbers can be thought of as skew-adjoint (since they are like 1-by-1 matrices), whereas real numbers correspond to self-adjoint operators.

Example

For example, the following matrix is skew-Hermitian:

{\begin{bmatrix}-i&2+i\\-(2-i)&0\end{bmatrix}}

Properties

The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.^[3]
All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, i.e., on the imaginary axis (the number zero is also considered purely imaginary).^[4]
If A, B are skew-Hermitian, then $aA+bB$ is skew-Hermitian for all real scalars a and b.^[5]
If A is skew-Hermitian, then both i A and −i A are Hermitian.^[5]
If A is skew-Hermitian, then A^k is Hermitian if k is an even integer and skew-Hermitian if k is an odd integer.
An arbitrary (square) matrix C can uniquely be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:^[2]

C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}(C+C^{\dagger })\quad {\mbox{and}}\quad B={\frac {1}{2}}(C-C^{\dagger }).

If A is skew-Hermitian, then e^A is unitary.
The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).

Notes

↑ Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
1 2 Horn & Johnson (1985), §4.1.2
↑ Horn & Johnson (1985), §2.5.2, §2.5.4
↑ Meyer (2000), Exercise 3.2.5
1 2 Horn & Johnson (1985), §4.1.1

References

Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 .
Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8 .

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[1] Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2

[HJ85S412-2] 1 2 Horn & Johnson (1985), §4.1.2

[3] Horn & Johnson (1985), §2.5.2, §2.5.4

[4] Meyer (2000), Exercise 3.2.5

[HJ85S411-5] 1 2 Horn & Johnson (1985), §4.1.1

Skew-Hermitian matrix

Example

Properties

See also

Notes

References