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Skew-Hermitian matrix

In linear algebra, a skew-Hermitian matrix is a square matrix with complex entries that equals the negative of its own conjugate transpose, denoted as A = -A^H, where A^H is the conjugate transpose of A.^[1] For real-valued matrices, this condition is equivalent to being skew-symmetric, meaning A = -A^T.^[1] These matrices form a real vector space, closed under addition and scalar multiplication by real numbers, and the diagonal entries of any skew-Hermitian matrix must be purely imaginary.^[2] Key properties of skew-Hermitian matrices include the fact that their eigenvalues are purely imaginary, lying on the imaginary axis of the complex plane.^[3] Moreover, they admit a spectral decomposition: any skew-Hermitian matrix A \in \mathbb{C}^{n \times n} can be unitarily diagonalized as A = U D U^H, where U is a unitary matrix and D is diagonal with purely imaginary entries.^[4] A notable relation to Hermitian matrices is that if A is skew-Hermitian, then iA is Hermitian, and conversely, multiplying a Hermitian matrix by i yields a skew-Hermitian one.^[2] Skew-Hermitian matrices play a fundamental role in the unique decomposition of any complex square matrix Z into a sum of a Hermitian matrix and a skew-Hermitian matrix: Z = \frac{1}{2}(Z + Z^H) + \frac{1}{2}(Z - Z^H), where the first term is Hermitian and the second is skew-Hermitian.^[3] They are also a subclass of normal matrices, satisfying A A^H = A^H A, which ensures the existence of an orthonormal basis of eigenvectors.^[5] In applications, such matrices arise in quantum mechanics for representing anti-Hermitian operators^[6] and in stability analysis of differential equations.^[7]

Definition

Formal Definition

A square matrix A with complex entries possesses a conjugate transpose, also known as the adjoint and denoted A^\dagger, which is obtained by first transposing A (replacing A with its transpose A^T, where (A^T)_{ij} = A_{ji}) and then taking the complex conjugate of each entry.^[8] The (i,j)-th entry of A^\dagger is thus given by (A^\dagger)_{ij} = \overline{A_{ji}}, where \overline{\cdot} denotes the complex conjugate.^[8] A skew-Hermitian matrix is a square matrix A over the complex numbers satisfying A^\dagger = -A.^[9] Skew-Hermitian matrices arise in the study of finite-dimensional complex vector spaces and are the complex analogs of real skew-symmetric matrices, which satisfy A^T = -A over the reals.^[9] They apply to n \times n matrices for any finite positive integer n.^[9] In contrast, Hermitian matrices satisfy A^\dagger = A.^[10]

Equivalent Conditions

A skew-Hermitian matrix A = (a_{ij}) satisfies the entry-wise condition that a_{ji} = -\overline{a_{ij}} for all indices i, j, where \overline{a_{ij}} denotes the complex conjugate of a_{ij}.^[1] This component-level characterization is equivalent to the defining property A^* = -A, with A^* being the conjugate transpose, and directly implies that the diagonal entries a_{ii} must be purely imaginary (or zero).^[1] From the perspective of inner product spaces, a skew-Hermitian matrix A represents a linear operator such that the associated sesquilinear form \langle Ax, y \rangle = -\langle x, Ay \rangle for all vectors x, y in the space, where \langle \cdot, \cdot \rangle is the standard inner product.^[11] This condition highlights the anti-symmetric nature of the form under the inner product, preserving a sesquilinear norm while introducing a sign reversal upon interchange of arguments.^[11] In a more general setting, skew-Hermitian matrices extend to bounded linear operators on complex Hilbert spaces, where an operator A is skew-Hermitian if its adjoint satisfies A^* = -A.^[12] This operator-theoretic formulation captures the same structural properties as the finite-dimensional matrix case, including the preservation of the Hilbert space inner product in an anti-Hermitian manner.^[12] A notable traceless variant consists of skew-Hermitian matrices with zero trace, which form the Lie algebra \mathfrak{su}(n) of the special unitary group \mathrm{SU}(n).^[13] These matrices underpin the infinitesimal generators of \mathrm{SU}(n) transformations and are essential in representation theory.^[13]

Properties

Basic Properties

The set of all n \times n skew-Hermitian matrices over the complex numbers forms a real vector space under the operations of matrix addition and scalar multiplication by real numbers. This structure arises because the defining condition A^\dagger = -A is preserved under these operations, making the set closed and satisfying the vector space axioms over the reals (though not over the complexes, as multiplication by i yields a Hermitian matrix).^[9] The collection is closed under addition: if A and B are skew-Hermitian, then (A + B)^\dagger = A^\dagger + B^\dagger = -A - B = -(A + B). Similarly, for any real scalar c, the multiple cA satisfies (cA)^\dagger = \overline{c} A^\dagger = c (-A) = -(cA), since \overline{c} = c. Regarding multiplication, the product AB of two skew-Hermitian matrices A and B is not generally skew-Hermitian; instead, (AB)^\dagger = B^\dagger A^\dagger = (-B)(-A) = BA, so AB is Hermitian precisely when A and B commute (i.e., AB = BA).^[14]^[15] The trace of a skew-Hermitian matrix A is purely imaginary (including zero). This follows from \operatorname{tr}(A^\dagger) = \overline{\operatorname{tr}(A)} = \operatorname{tr}(-A) = -\operatorname{tr}(A), implying $2 \operatorname{Re}(\operatorname{tr}(A)) = 0, so the real part vanishes. Additionally, every skew-Hermitian matrix is normal, as A A^\dagger = A (-A) = -A^2 and A^\dagger A = (-A) A = -A^2, confirming A A^\dagger = A^\dagger A.^[16]^[17]

Spectral Properties

A skew-Hermitian matrix has all eigenvalues purely imaginary. To see this, suppose A is skew-Hermitian, so A^\dagger = -A, and let \lambda be an eigenvalue with eigenvector v \neq 0, satisfying A v = \lambda v. Taking the conjugate transpose gives v^\dagger A^\dagger = \overline{\lambda} v^\dagger, or v^\dagger (-A) = \overline{\lambda} v^\dagger, hence -v^\dagger A = \overline{\lambda} v^\dagger. Multiplying on the right by v yields -v^\dagger A v = \overline{\lambda} v^\dagger v. But v^\dagger A v = v^\dagger (\lambda v) = \lambda v^\dagger v, so -\lambda \|v\|^2 = \overline{\lambda} \|v\|^2, implying \lambda = -\overline{\lambda}, or equivalently, the real part of \lambda is zero. Thus, each eigenvalue can be written as \lambda = i \mu where \mu is real.^[16]^[18] Skew-Hermitian matrices are normal, since A A^\dagger = A (-A) = -A^2 and A^\dagger A = (-A) A = -A^2, so A A^\dagger = A^\dagger A. Normal matrices over the complex numbers are unitarily diagonalizable, meaning there exists a unitary matrix U (satisfying U^\dagger U = I) such that A = U D U^\dagger, where D is diagonal with purely imaginary entries on the diagonal. The columns of U form an orthonormal basis of eigenvectors for A, and eigenvectors corresponding to distinct eigenvalues are orthogonal.^[19]^[18] Since skew-Hermitian matrices are normal, they are diagonalizable and possess only trivial Jordan blocks; their Jordan canonical form is simply the diagonal matrix D from the unitary diagonalization, with no off-diagonal 1's. The determinant of a skew-Hermitian matrix is the product of its eigenvalues, hence a product of purely imaginary numbers.^[20]^[18]

Examples and Constructions

Explicit Examples

A skew-Hermitian matrix of order 1 is simply a scalar multiple of the imaginary unit, such as [i b] where b is real, since its conjugate transpose is [-i b], which equals the negative only if b is real.^[21] For a general 2×2 skew-Hermitian matrix, the form is

A = \begin{pmatrix} i a & b + i c \\ -b + i c & i d \end{pmatrix},

where a, d are real and b, c are real numbers, ensuring the diagonal entries are purely imaginary and the off-diagonal entries satisfy the skew-Hermitian condition A^\dagger = -A. To verify, the conjugate transpose yields

A^\dagger = \begin{pmatrix} -i a & -b - i c \\ b - i c & -i d \end{pmatrix} = -A.

^[21] A specific numerical example is the matrix

\begin{pmatrix} -i & 2 - i \\ -2 - i & 0 \end{pmatrix}.

Its conjugate transpose is

\begin{pmatrix} i & -2 + i \\ 2 + i & 0 \end{pmatrix},

which equals the negative of the original matrix, confirming it is skew-Hermitian.^[21] Over the real numbers, skew-Hermitian matrices coincide with skew-symmetric matrices, such as

\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},

whose transpose is its negative.^[21] Traceless skew-Hermitian matrices form the Lie algebra \mathfrak{su}(2), with basis given by i times the Pauli matrices:

i \sigma_x = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \quad i \sigma_y = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad i \sigma_z = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}.

Each satisfies the skew-Hermitian condition and has trace zero.^[22]

Generating Methods

A fundamental method to generate a skew-Hermitian matrix involves multiplying a Hermitian matrix by the imaginary unit i. If H is Hermitian, satisfying H^\dagger = H, then the matrix S = i H is skew-Hermitian because S^\dagger = (i H)^\dagger = -i H^\dagger = -i H = -S.^[17] For direct construction of finite-dimensional skew-Hermitian matrices, one specifies arbitrary complex entries above the main diagonal, sets the diagonal entries to purely imaginary numbers (including zero), and defines the entries below the main diagonal as the negative complex conjugates of the corresponding upper triangular entries. This procedure ensures the resulting matrix S satisfies S^\dagger = -S by design, as the conjugate transpose swaps and conjugates the off-diagonal elements while negating the diagonal imaginaries appropriately. In numerical and computational contexts, random skew-Hermitian matrices are often generated by first creating a random Hermitian matrix—typically by drawing independent complex Gaussian entries for the upper triangle, their conjugates for the lower triangle, and real Gaussians for the diagonal—and then multiplying by i. This approach leverages efficient algorithms for random Hermitian generation and preserves the skew-Hermitian structure probabilistically. Skew-Hermitian matrices can also be constructed in block forms. A block-diagonal matrix composed of skew-Hermitian blocks on the diagonal is itself skew-Hermitian, since the conjugate transpose acts block-wise and preserves the property for each component. Similarly, the Kronecker product of a skew-Hermitian matrix A and a Hermitian matrix B yields a skew-Hermitian matrix, as (A \otimes B)^\dagger = A^\dagger \otimes B^\dagger = (-A) \otimes B = -(A \otimes B). In infinite-dimensional settings, skew-Hermitian operators arise naturally from differential operators on Hilbert spaces such as L^2(\mathbb{R}). For instance, the differential operator i \frac{d}{dx} (with suitable domain, e.g., smooth compactly supported functions) defines a skew-Hermitian operator, as its formal adjoint is -i \frac{d}{dx}, satisfying the required relation under appropriate boundary conditions.

Decompositions and Representations

Hermitian-Skew-Hermitian Decomposition

Every square complex matrix A admits a unique decomposition into a Hermitian part and a skew-Hermitian part, given by

A = H + S,

where

H = \frac{A + A^\dagger}{2}, \quad S = \frac{A - A^\dagger}{2}.

Here, A^\dagger denotes the conjugate transpose (adjoint) of A, H satisfies H^\dagger = H, and S satisfies S^\dagger = -S. To verify the properties, compute

H^\dagger = \left( \frac{A + A^\dagger}{2} \right)^\dagger = \frac{A^\dagger + A}{2} = H,

S^\dagger = \left( \frac{A - A^\dagger}{2} \right)^\dagger = \frac{A^\dagger - A}{2} = -\frac{A - A^\dagger}{2} = -S,

and

H + S = \frac{A + A^\dagger + A - A^\dagger}{2} = A.

This establishes the decomposition. The decomposition is unique: Suppose A = H' + S' with H'^\dagger = H' and (S')^\dagger = -S'. Then H' - H = S - S'. Let D = H' - H = S - S'. The left side yields D^\dagger = D, while the right side yields D^\dagger = -D, so D = -D, implying D = 0. Thus, H' = H and S' = S. This decomposition, known as the Toeplitz or Cartesian decomposition, extends to bounded linear operators on complex Hilbert spaces, where any T \in B(\mathcal{H}) satisfies T = \operatorname{Re}(T) + i \operatorname{Im}(T), with both \operatorname{Re}(T) and \operatorname{Im}(T) self-adjoint (corresponding to Hermitian and skew-Hermitian parts via S = i \operatorname{Im}(T)). In numerical analysis, the decomposition enhances stability in iterative solvers for non-Hermitian systems, such as Hermitian and skew-Hermitian splitting (HSS) methods for large sparse positive definite linear equations, and supports computations like polar decompositions by isolating symmetric components.

Exponential Map

The matrix exponential of a skew-Hermitian matrix S, defined as \exp(S) = \sum_{k=0}^{\infty} \frac{S^k}{k!}, is always a unitary matrix. This follows from the property that S^\dagger = -S, which implies \exp(S)^\dagger = \exp(S^\dagger) = \exp(-S) = \exp(S)^{-1}, confirming the unitarity condition \exp(S)^\dagger \exp(S) = I. The power series for the matrix exponential converges absolutely for every skew-Hermitian matrix S, as it does for all complex square matrices, with the radius of convergence being infinite. This convergence ensures that \exp(S) is well-defined and analytic in the space of skew-Hermitian matrices. A closely related construction involves Hermitian matrices: for any Hermitian matrix H, the matrix \exp(iH) is unitary, where iH is skew-Hermitian. Conversely, every exponential of a skew-Hermitian matrix can be expressed in this form by setting S = iH with H = -iS Hermitian. This equivalence highlights the exponential map's role in parameterizing unitary matrices via skew-Hermitian arguments. Every unitary matrix U admits a skew-Hermitian logarithm \Log(U) such that \exp(\Log(U)) = U. The principal branch of this logarithm is uniquely defined as the one whose eigenvalues lie in the open horizontal strip \{ z \in \mathbb{C} : -\pi < \Im(z) \leq \pi \}, ensuring the eigenvalues of \Log(U) are purely imaginary within i(-\pi, \pi]. Since skew-Hermitian matrices are normal and thus unitarily diagonalizable, the exponential can be computed efficiently via spectral decomposition: if S = V D V^\dagger with V unitary and D diagonal containing purely imaginary entries, then \exp(S) = V \exp(D) V^\dagger, where \exp(D) applies the scalar exponential entrywise. This method leverages the unitary diagonalization property of skew-Hermitian matrices.

Applications

Lie Groups and Algebras

The set of all n \times n skew-Hermitian matrices over the complex numbers forms the Lie algebra \mathfrak{u}(n) of the unitary group U(n), where the Lie bracket is the commutator [A, B] = AB - BA.^[23] This structure bridges linear algebra with differential geometry, as \mathfrak{u}(n) consists precisely of the tangent vectors at the identity element of U(n), which is the Lie group of all n \times n unitary matrices preserving the standard inner product on \mathbb{C}^n.^[13] The real dimension of \mathfrak{u}(n) is n^2, arising from the fact that a basis for \mathfrak{u}(n) can be constructed as i times a basis for the real vector space of n \times n Hermitian matrices, which also has dimension n^2.^[24] The subalgebra \mathfrak{su}(n) of traceless skew-Hermitian matrices is the Lie algebra of the special unitary group SU(n), the kernel of the determinant map U(n) \to S^1, and possesses real dimension n^2 - 1.^[25] The adjoint representation of U(n) on \mathfrak{u}(n) is realized by conjugation: for U \in U(n) and A \in \mathfrak{u}(n), \mathrm{Ad}_U(A) = U A U^{-1}, preserving the Lie bracket and thus the algebra structure.^[13] The exponential map \exp: \mathfrak{u}(n) \to U(n), given by the matrix exponential series, induces a local diffeomorphism near the zero element, mapping \mathfrak{u}(n) onto U(n) and reflecting the covering of the group by its algebra in the infinitesimal sense.^[26]

Physics and Engineering

In quantum mechanics, skew-Hermitian matrices play a fundamental role in describing the time evolution of quantum states. The time evolution operator U(t) = \exp\left( -\frac{i H t}{\hbar} \right), where H is the Hermitian Hamiltonian operator representing the system's energy, generates unitary dynamics that preserve the norm of the state vector. Here, the argument K = -\frac{i H t}{\hbar} is skew-Hermitian, as its eigenvalues are purely imaginary, ensuring that U(t) remains unitary and the evolution is reversible. This structure underpins the Schrödinger equation i \hbar \frac{d}{dt} |\psi(t)\rangle = H |\psi(t)\rangle, where the skew-Hermitian generator maintains probability conservation.^[27] In quantum computing, skew-Hermitian matrices appear as generators in variational quantum algorithms (VQAs) and simulation techniques. For instance, in the variational quantum eigensolver (VQE), the ansatz circuits are constructed from exponentials of skew-Hermitian operators derived from Hermitian Pauli strings, enabling optimization over unitary transformations to approximate ground states of complex Hamiltonians. Similarly, the Lie-Trotter decomposition approximates the exponential of a sum of skew-Hermitian terms for efficient simulation of quantum dynamics on near-term devices, reducing circuit depth while preserving unitarity. These methods leverage the skew-Hermitian property to ensure faithful representation of quantum evolutions in noisy intermediate-scale quantum hardware.^[28]^[29] Skew-Hermitian matrices are integral to signal processing applications, particularly in the design of paraunitary filter banks for perfect reconstruction. Paraunitary systems, which maintain energy preservation across subbands, are constructed via the Cayley transform that maps para-skew-Hermitian matrices to paraunitary ones, converting nonlinear constraints into linear equations for easier optimization. The exponential of a skew-Hermitian matrix \exp(S), being unitary, ensures that the filter bank polyphase matrix satisfies the paraunitary condition P(z) \tilde{P}(z) = I, enabling lossless decomposition in applications like image compression and audio coding. This approach simplifies the synthesis of compactly supported orthogonal wavelets with desirable frequency selectivity.^[30]^[31] In numerical methods for optimization, skew-Hermitian approximations facilitate gradient flows on unitary manifolds, which are prevalent in machine learning tasks with orthogonal constraints. The tangent space to the unitary group consists of skew-Hermitian matrices, allowing Riemannian gradient descent to project updates onto this space for maintaining unitarity during training of recurrent neural networks or quantum circuit parameters. For example, in optimizing over the special unitary group, the skew-Hermitian component of the Euclidean gradient is extracted to define the Riemannian metric, enabling convergence to local minima while avoiding manifold violations. This is particularly effective in quantum machine learning, where it approximates complex evolutions with reduced computational overhead.^[32]^[33]

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### Definition and Formula for the Adjoint (Conjugate Transpose) of a Matrix
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Antihermitian matrices are often called "skew Hermitian matrices" by mathematicians. ... See also. Adjoint, Antisymmetric Part, Hermitian Matrix, Unitary Matrix.
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### Definition of Hermitian Matrix
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hence the Lie algebra su(2) of SU(2) consists of all traceless two-by-two skew-hermitian matrices: su(2) = {X ∈ Mat(2, C) : X = −X†, tr X = 0}. A basis for ...
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