Skew-Hermitian matrix

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

${\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad A^{\mathsf {H}}=-A}$

where ${\displaystyle A^{\textsf {H}}}$ denotes the conjugate transpose of the matrix ${\displaystyle A}$. In component form, this means that

${\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad a_{ij}=-{\overline {a_{ji}}}}$

for all indices ${\displaystyle i}$ and ${\displaystyle j}$, where ${\displaystyle a_{ij}}$ is the element in the ${\displaystyle i}$-th row and ${\displaystyle j}$-th column of ${\displaystyle A}$, 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 ${\displaystyle n\times n}$ matrices forms the ${\displaystyle u(n)}$ 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 ${\displaystyle n}$ dimensional complex or real space ${\displaystyle K^{n}}$. If ${\displaystyle (\cdot \mid \cdot )}$ denotes the scalar product on ${\displaystyle K^{n}}$, then saying ${\displaystyle A}$ is skew-adjoint means that for all ${\displaystyle \mathbf {u} ,\mathbf {v} \in K^{n}}$ one has ${\displaystyle (A\mathbf {u} \mid \mathbf {v} )=-(\mathbf {u} \mid A\mathbf {v} )}$.

Imaginary numbers can be thought of as skew-adjoint (since they are like ${\displaystyle 1\times 1}$ matrices), whereas real numbers correspond to self-adjoint operators.

Example

For example, the following matrix is skew-Hermitian

$${\displaystyle A={\begin{bmatrix}-i&+2+i\\-2+i&0\end{bmatrix}}}$$

because

$${\displaystyle -A={\begin{bmatrix}i&-2-i\\2-i&0\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {-2+i}}\\{\overline {2+i}}&{\overline {0}}\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {2+i}}\\{\overline {-2+i}}&{\overline {0}}\end{bmatrix}}^{\mathsf {T}}=A^{\mathsf {H}}}$$

Properties

• The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly 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 ${\displaystyle A}$ and ${\displaystyle B}$ are skew-Hermitian, then ${\displaystyle aA+bB}$ is skew-Hermitian for all real scalars ${\displaystyle a}$ and ${\displaystyle b}$. ^[5]
• ${\displaystyle A}$ is skew-Hermitian if and only if${\displaystyle iA}$ (or equivalently, ${\displaystyle -iA}$) is Hermitian. ^[5]
• ${\displaystyle A}$ is skew-Hermitian if and only if the real part ${\displaystyle \Re {(A)}}$ is skew-symmetric and the imaginary part ${\displaystyle \Im {(A)}}$ is symmetric.
• If ${\displaystyle A}$ is skew-Hermitian, then ${\displaystyle A^{k}}$ is Hermitian if ${\displaystyle k}$ is an even integer and skew-Hermitian if ${\displaystyle k}$ is an odd integer.
• ${\displaystyle A}$ is skew-Hermitian if and only if ${\displaystyle \mathbf {x} ^{\mathsf {H}}A\mathbf {y} =-\mathbf {y} ^{\mathsf {H}}A\mathbf {x} }$ for all vectors ${\displaystyle \mathbf {x} ,\mathbf {y} }$.
• If ${\displaystyle A}$ is skew-Hermitian, then the matrix exponential ${\displaystyle e^{A}}$ is unitary.
• The space of skew-Hermitian matrices forms the Lie algebra ${\displaystyle u(n)}$ of the Lie group ${\displaystyle U(n)}$.

Decomposition into Hermitian and skew-Hermitian

• The sum of a square matrix and its conjugate transpose ${\displaystyle \left(A+A^{\mathsf {H}}\right)}$ is Hermitian.
• The difference of a square matrix and its conjugate transpose ${\displaystyle \left(A-A^{\mathsf {H}}\right)}$ is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
• An arbitrary square matrix ${\displaystyle C}$ can be written as the sum of a Hermitian matrix ${\displaystyle A}$ and a skew-Hermitian matrix ${\displaystyle B}$:
  $${\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}$$

See also

• Bivector (complex)
• Hermitian matrix
• Normal matrix
• Skew-symmetric matrix
• Unitary matrix

Notes

1. Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
2. Horn & Johnson (1985), §4.1.2
3. Horn & Johnson (1985), §2.5.2, §2.5.4
4. Meyer (2000), Exercise 3.2.5
5. 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 .