Skew-Hermitian matrix
Definition and Properties
A skew-Hermitian matrix (also known as an anti-Hermitian matrix) is a square matrix \( A \) with complex entries that satisfies the condition \( A^* = -A \), where \( A^* \) denotes the conjugate transpose of \( A \). In other words, a matrix \( A \) is skew-Hermitian if and only if \( A_{ij} = -\overline{A_{ji}} \) for all \( i \) and \( j \), where \( \overline{A_{ji}} \) represents the complex conjugate of the entry \( A_{ji} \).
Skew-Hermitian matrices are a generalization of skew-symmetric matrices to the field of complex numbers. They play a significant role in various branches of mathematics and physics, particularly in linear algebra, quantum mechanics, and theory of Lie groups.
Mathematical Formulation
Let \( A \) be an \( n \times n \) skew-Hermitian matrix. Then, by definition: \[ A^* = -A \] This implies that the diagonal elements of \( A \) must be purely imaginary or zero, as \( A_{ii} = -\overline{A_{ii}} \) leads to \( A_{ii} + \overline{A_{ii}} = 0 \), which means \( 2\Re(A_{ii}) = 0 \), where \( \Re(A_{ii}) \) denotes the real part of \( A_{ii} \).
For the off-diagonal elements, the condition \( A_{ij} = -\overline{A_{ji}} \) must hold. This property ensures that the matrix is not only skew-Hermitian but also that its structure is inherently tied to the properties of complex conjugation.
Eigenvalues and Eigenvectors
The eigenvalues of a skew-Hermitian matrix are purely imaginary or zero. To see why this is the case, consider an eigenvalue \( \lambda \) and an eigenvector \( \mathbf{v} \) such that: \[ A\mathbf{v} = \lambda\mathbf{v} \] Taking the conjugate transpose of both sides, we get: \[ \mathbf{v}^* A^* = \overline{\lambda} \mathbf{v}^* \] Since \( A^* = -A \), this becomes: \[ \mathbf{v}^* (-A) = \overline{\lambda} \mathbf{v}^* \] \[ -\mathbf{v}^* A = \overline{\lambda} \mathbf{v}^* \] Multiplying both sides by \( -1 \), we obtain: \[ \mathbf{v}^* A = -\overline{\lambda} \mathbf{v}^* \] Comparing this with the original eigenvalue equation, we see that: \[ \lambda = -\overline{\lambda} \] This implies that \( \lambda \) is purely imaginary or zero.
Applications
Skew-Hermitian matrices are crucial in various applications, including:
Quantum Mechanics
In quantum mechanics, skew-Hermitian matrices are often encountered in the context of Hamiltonians and observable operators. The commutation relations between operators can be represented using skew-Hermitian matrices.
Lie Groups and Lie Algebras
Skew-Hermitian matrices form the Lie algebra of the unitary group \( U(n) \). The set of all \( n \times n \) skew-Hermitian matrices, with the commutator as the Lie bracket, forms a Lie algebra denoted by \( \mathfrak{u}(n) \). This algebra is fundamental in the study of Lie groups and their representations.
Control Theory
In control theory, skew-Hermitian matrices are used in the analysis and design of linear systems. They appear in the context of stability analysis and in the formulation of certain types of differential equations.
Examples
Consider the following \( 2 \times 2 \) skew-Hermitian matrix: \[ A = \begin{pmatrix} i & 1+i \\ -1+i & -i \end{pmatrix} \] Here, \( A^* = \begin{pmatrix} -i & -1-i \\ 1-i & i \end{pmatrix} \), and it can be verified that \( A^* = -A \).
Another example is the \( 3 \times 3 \) skew-Hermitian matrix: \[ B = \begin{pmatrix} 0 & -i & 2i \\ i & 0 & -3 \\ -2i & 3 & 0 \end{pmatrix} \] For this matrix, \( B^* = \begin{pmatrix} 0 & i & -2i \\ -i & 0 & 3 \\ 2i & -3 & 0 \end{pmatrix} \), and indeed \( B^* = -B \).
Properties
Norms and Inner Products
For any skew-Hermitian matrix \( A \), the Frobenius norm \( \|A\|_F \) is defined as: \[ \|A\|_F = \sqrt{\sum_{i,j} |A_{ij}|^2} \] This norm is invariant under unitary transformations. Moreover, the inner product of two skew-Hermitian matrices \( A \) and \( B \) in the space of \( n \times n \) matrices is given by: \[ \langle A, B \rangle = \text{Tr}(A^* B) \] where \( \text{Tr} \) denotes the trace of the matrix.
Decomposition
Any complex matrix \( M \) can be decomposed into a Hermitian part and a skew-Hermitian part. Specifically: \[ M = H + S \] where \( H = \frac{M + M^*}{2} \) is Hermitian and \( S = \frac{M - M^*}{2} \) is skew-Hermitian. This decomposition is unique and is useful in various mathematical and physical contexts.
Commutativity and Anticommutativity
If \( A \) and \( B \) are skew-Hermitian matrices, their commutator \( [A, B] = AB - BA \) is also skew-Hermitian. This property is significant in the study of Lie algebras. Additionally, the anticommutator \( \{A, B\} = AB + BA \) is Hermitian.