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. That is, the matrix A {\displaystyle A} is skew-Hermitian if it satisfies the relation
A skew-Hermitian ⟺ A H = − A {\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad A^{\mathsf {H}}=-A}
where A H {\displaystyle A^{\textsf {H}}} denotes the conjugate transpose of the matrix A {\displaystyle A} . In component form, this means that
A skew-Hermitian ⟺ a i j = − a j i ¯ {\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad a_{ij}=-{\overline {a_{ji}}}}
for all indices i {\displaystyle i} and j {\displaystyle j} , where a i j {\displaystyle a_{ij}} is the element in the i {\displaystyle i} -th row and j {\displaystyle j} -th column of A {\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. The set of all skew-Hermitian n × n {\displaystyle n\times n} matrices forms the u ( n ) {\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 n {\displaystyle n} dimensional complex or real space K n {\displaystyle K^{n}} . If ( ⋅ ∣ ⋅ ) {\displaystyle (\cdot \mid \cdot )} denotes the scalar product on K n {\displaystyle K^{n}} , then saying A {\displaystyle A} is skew-adjoint means that for all u , v ∈ K n {\displaystyle \mathbf {u} ,\mathbf {v} \in K^{n}} one has ( A u ∣ v ) = − ( u ∣ A v ) {\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 1 × 1 {\displaystyle 1\times 1} matrices), whereas real numbers correspond to self-adjoint operators.
Example
For example, the following matrix is skew-Hermitian A = [ − i + 2 + i − 2 + i 0 ] {\displaystyle A={\begin{bmatrix}-i&+2+i\\-2+i&0\end{bmatrix}}} because − A = [ i − 2 − i 2 − i 0 ] = [ − i ¯ − 2 + i ¯ 2 + i ¯ 0 ¯ ] = [ − i ¯ 2 + i ¯ − 2 + i ¯ 0 ¯ ] T = A H {\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 A {\displaystyle A} and B {\displaystyle B} are skew-Hermitian, then a A + b B {\displaystyle aA+bB} is skew-Hermitian for all real scalars a {\displaystyle a} and b {\displaystyle b} .5
- A {\displaystyle A} is skew-Hermitian if and only if i A {\displaystyle iA} (or equivalently, − i A {\displaystyle -iA} ) is Hermitian.6
- A {\displaystyle A} is skew-Hermitian if and only if the real part ℜ ( A ) {\displaystyle \Re {(A)}} is skew-symmetric and the imaginary part ℑ ( A ) {\displaystyle \Im {(A)}} is symmetric.
- If A {\displaystyle A} is skew-Hermitian, then A k {\displaystyle A^{k}} is Hermitian if k {\displaystyle k} is an even integer and skew-Hermitian if k {\displaystyle k} is an odd integer.
- A {\displaystyle A} is skew-Hermitian if and only if x H A y = − y H A x ¯ {\displaystyle \mathbf {x} ^{\mathsf {H}}A\mathbf {y} =-{\overline {\mathbf {y} ^{\mathsf {H}}A\mathbf {x} }}} for all vectors x , y {\displaystyle \mathbf {x} ,\mathbf {y} } .
- If A {\displaystyle A} is skew-Hermitian, then the matrix exponential e A {\displaystyle e^{A}} is unitary.
- The space of skew-Hermitian matrices forms the Lie algebra u ( n ) {\displaystyle u(n)} of the Lie group U ( n ) {\displaystyle U(n)} .
Decomposition into Hermitian and skew-Hermitian
- The sum of a square matrix and its conjugate transpose ( A + A H ) {\displaystyle \left(A+A^{\mathsf {H}}\right)} is Hermitian.
- The difference of a square matrix and its conjugate transpose ( A − A H ) {\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 C {\displaystyle C} can be written as the sum of a Hermitian matrix A {\displaystyle A} and a skew-Hermitian matrix B {\displaystyle B} : C = A + B with A = 1 2 ( C + C H ) and B = 1 2 ( C − C H ) {\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
Notes
- 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.
References
Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2 - Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 ↩
Horn & Johnson (1985), §4.1.2 - Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 ↩
Horn & Johnson (1985), §2.5.2, §2.5.4 - Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 ↩
Meyer (2000), Exercise 3.2.5 - Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8 http://www.matrixanalysis.com/ ↩
Horn & Johnson (1985), §4.1.1 - Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 ↩
Horn & Johnson (1985), §4.1.1 - Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 ↩