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Skew-symmetric matrix

In linear algebra, a skew-symmetric matrix (also called an antisymmetric matrix) is a square matrix A satisfying A^T = -A, which implies that its elements satisfy a_{ij} = -a_{ji} for all indices i and j.^[1] This condition ensures that the matrix is equal to the negative of its transpose, distinguishing it from symmetric matrices where A^T = A.^[2] Skew-symmetric matrices form a vector subspace of the space of all square matrices, closed under addition and scalar multiplication, and any real square matrix can be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.^[1]^[3] Key properties of skew-symmetric matrices include zero diagonal elements, as a_{ii} = -a_{ii} forces a_{ii} = 0 for all i, making the matrix hollow.^[1] For real skew-symmetric matrices, all eigenvalues are either zero or purely imaginary, and they occur in complex conjugate pairs; moreover, the matrix is singular if its dimension is odd.^[4] Over the real numbers, the rank of an n \times n skew-symmetric matrix is always even.^[5] These matrices are also normal (commuting with their adjoint) when considered over the complexes, and their exponentials yield orthogonal matrices with determinant 1, connecting them to the special orthogonal group SO(n).^[5] Skew-symmetric matrices play a fundamental role in geometry and physics, particularly in representing infinitesimal rotations and the Lie algebra of the rotation group SO(3), where elements correspond to angular velocities or infinitesimal rigid body motions.^[6] In quantum mechanics and classical mechanics, they model angular momentum operators and cross-product structures, facilitating the study of conservation laws and symmetries.^[7] Their pfaffian invariant further links them to determinants and combinatorial problems in even dimensions.^[8]

Definition and Basic Concepts

Definition

In linear algebra, a skew-symmetric matrix is a square matrix A over the real numbers satisfying A^T = -A, where A^T denotes the transpose of A.^[9] This condition implies that the entries of the matrix obey a_{ij} = -a_{ji} for all indices i and j, with the diagonal entries necessarily zero since a_{ii} = -a_{ii} forces a_{ii} = 0.^[9]^[10] The definition extends to square matrices over any field of characteristic not equal to 2, where the relation A^T = -A remains meaningful and distinct from the symmetric case A^T = A.^[10] In fields of characteristic 2, however, the condition A^T = -A simplifies to A^T = A because -1 = 1, causing skew-symmetric matrices to coincide with symmetric ones and rendering the notion less useful without additional structure like alternating forms.^[11] Skew-symmetric matrices serve as the antipodal counterpart to symmetric matrices in the decomposition of arbitrary square matrices.^[9]

Elementary Properties

A skew-symmetric matrix A satisfies A^T = -A, which immediately implies that all diagonal entries are zero. For the (i,i)-th entry, a_{ii} = -a_{ii}, so $2a_{ii} = 0 and thus a_{ii} = 0.^[9] The trace of A, defined as the sum of its diagonal entries \operatorname{tr}(A) = \sum_{i} a_{ii}, is therefore zero.^[12] The set of skew-symmetric matrices is closed under addition and scalar multiplication. If A and B are skew-symmetric, then (A + B)^T = A^T + B^T = -A - B = -(A + B), so A + B is skew-symmetric.^[13] Similarly, for any scalar c, (cA)^T = c A^T = c (-A) = - (cA), confirming that cA is skew-symmetric.^[14] The product of two skew-symmetric matrices need not be skew-symmetric. If A and B are skew-symmetric, then (AB)^T = B^T A^T = (-B)(-A) = BA, which equals -AB only if AB = -BA. However, the square A^2 of a skew-symmetric matrix is symmetric: (A^2)^T = (A^T)^2 = (-A)^2 = A^2.^[15] For real matrices, the skew-symmetric condition A^T = -A coincides with the skew-Hermitian condition A^* = -A, where A^* denotes the conjugate transpose, since conjugation has no effect on real entries.^[16]

Examples

Low-Dimensional Cases

In the one-dimensional case, the only skew-symmetric matrix is the zero matrix

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, since the single diagonal entry a must satisfy a = -a, which implies a = 0.^[17] For two dimensions, a general real skew-symmetric $2 \times 2 matrix takes the form

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

where a \in \mathbb{R} is the single independent parameter above the diagonal.^[18] An example is

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

which generates a scaled 90-degree rotation when exponentiated.^[19] In three dimensions, a general real skew-symmetric $3 \times 3 matrix has three independent parameters and takes the form

\begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix},

where a, b, c \in \mathbb{R}.^[20] A concrete example is the matrix associated with the cross product operation for the vector \mathbf{v} = (x, y, z), given by

[\mathbf{v}]_\times = \begin{pmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{pmatrix},

which satisfies the skew-symmetric condition and maps vectors via matrix-vector multiplication to mimic the cross product.^[20] Skew-symmetric matrices in low dimensions can be interpreted as representing antisymmetric bilinear forms on vectors: for a matrix A, the form B(\mathbf{u}, \mathbf{v}) = \mathbf{u}^T A \mathbf{v} satisfies B(\mathbf{u}, \mathbf{v}) = -B(\mathbf{v}, \mathbf{u}).^[21] All such matrices have zero trace, as their diagonal entries are necessarily zero.^[17]

General Constructions

A skew-symmetric matrix of size n \times n has exactly \frac{n(n-1)}{2} independent entries, as the diagonal elements must be zero and the entries below the diagonal are determined by those above it via the relation a_{ji} = -a_{ij}. One general method to construct larger skew-symmetric matrices is through the direct sum (or block-diagonal form) of smaller skew-symmetric matrices. If A and B are skew-symmetric matrices of sizes p \times p and q \times q, respectively, their direct sum A \oplus B is the (p+q) \times (p+q) block-diagonal matrix with A and B on the diagonal blocks and zeros elsewhere; this is skew-symmetric because (A \oplus B)^T = A^T \oplus B^T = (-A) \oplus (-B) = -(A \oplus B). Low-dimensional skew-symmetric matrices can serve as building blocks in such constructions. In three dimensions, a skew-symmetric $3 \times 3 matrix can be constructed from a vector \mathbf{v} = (v_1, v_2, v_3) as

[\mathbf{v}]_\times = \begin{pmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{pmatrix},

which satisfies [\mathbf{v}]_\times^T = -[\mathbf{v}]_\times. The matrix exponential provides another construction route: for a skew-symmetric matrix A, the matrix e^{tA} is orthogonal for any real scalar t, since (e^{tA})^T = e^{tA^T} = e^{-tA} = (e^{tA})^{-1}.^[22]

Algebraic Properties

Vector Space Structure

The set of all n \times n skew-symmetric matrices over the real numbers \mathbb{R} forms a vector space under matrix addition and scalar multiplication.^[23] This space is closed under these operations: if A^T = -A and B^T = -B, then (aA + bB)^T = aA^T + bB^T = -aA - bB = -(aA + bB) for any scalars a, b \in \mathbb{R}.^[24] The dimension of this vector space is n(n-1)/2, corresponding to the number of independent entries above the main diagonal, as the diagonal elements must be zero and the lower triangle is determined by the upper triangle via the skew-symmetry condition.^[23] A standard basis for this vector space consists of the matrices E_{ij} - E_{ji} for $1 \leq i < j \leq n, where E_{kl} denotes the n \times n matrix with a 1 in position (k,l) and zeros elsewhere.^[25] These basis elements are linearly independent and span the space, as any skew-symmetric matrix A = [a_{kl}] can be uniquely expressed as A = \sum_{1 \leq i < j \leq n} a_{ij} (E_{ij} - E_{ji}).^[25] The space of skew-symmetric matrices admits the Frobenius inner product, defined for any two n \times n matrices A, B by \langle A, B \rangle = \operatorname{tr}(A^T B) = \sum_{i,j=1}^n a_{ij} b_{ij}, where \operatorname{tr} denotes the trace.^[24] For skew-symmetric matrices A and B, this simplifies to \langle A, B \rangle = \operatorname{tr}(A^T B) = \operatorname{tr}((-A) B) = -\operatorname{tr}(A B), reflecting the anti-symmetric nature of the pairing.^[24] This inner product establishes orthogonality between the space of skew-symmetric matrices and the space of symmetric matrices: if A is skew-symmetric and S is symmetric (S^T = S), then \langle A, S \rangle = \operatorname{tr}(A^T S) = \operatorname{tr}((-A) S) = -\operatorname{tr}(A S). But \operatorname{tr}(A S) = \operatorname{tr}(S A) = \operatorname{tr}((A S)^T) = \operatorname{tr}(S^T A^T) = \operatorname{tr}(S (-A)) = -\operatorname{tr}(S A) = -\operatorname{tr}(A S), implying \operatorname{tr}(A S) = 0 and thus \langle A, S \rangle = 0.^[26] This decomposition highlights how the full matrix space \mathbb{R}^{n \times n} splits into orthogonal subspaces of symmetric and skew-symmetric matrices under the Frobenius inner product.^[26]

Determinant and Pfaffian

For a skew-symmetric matrix A of odd order n, the determinant is always zero. This follows from the property that \det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A); since n is odd, \det(A) = -\det(A), implying \det(A) = 0. Thus, such matrices are singular.^[27] For a real skew-symmetric matrix A of even order n = 2m, the determinant is non-negative and equals the square of the Pfaffian: \det(A) = \Pf(A)^2 \geq 0. The Pfaffian \Pf(A) is defined for such matrices as the sum over all perfect matchings \sigma of \{1, 2, \dots, 2m\}, where each matching pairs the indices into m disjoint transpositions (i_1 j_1), \dots, (i_m j_m) with i_k < j_k and i_1 < i_2 < \dots < i_m:

\Pf(A) = \sum_{\sigma} \operatorname{sgn}(\sigma) \prod_{k=1}^m a_{i_k j_k},

with \operatorname{sgn}(\sigma) the sign of the corresponding permutation. This polynomial in the entries of A satisfies the square relation, providing a square root of the determinant.^[27]^[28] The Pfaffian was introduced by Arthur Cayley in 1852 and named after the German mathematician Johann Friedrich Pfaff, who studied related systems of differential equations in 1814–1815. It has applications in statistical mechanics, such as computing partition functions for the Ising model on planar graphs via the Pfaffian of an associated skew-symmetric matrix.^[29]^[30]

Spectral Theory

For a real skew-symmetric matrix A \in \mathbb{R}^{n \times n}, the eigenvalues are either zero or purely imaginary, occurring in conjugate pairs \pm i \lambda where \lambda > 0. This property arises because iA is symmetric (hence Hermitian over \mathbb{C}), and the spectral theorem for Hermitian matrices guarantees real eigenvalues for iA, implying pure imaginary eigenvalues for A.^[31]^[32] The characteristic polynomial p(\lambda) = \det(\lambda I - A) is an even function, satisfying p(\lambda) = p(-\lambda). This follows from the relation \det(\lambda I - A) = \det((\lambda I - A)^T) = \det(\lambda I + A), reflecting the skew-symmetry. Consequently, non-zero eigenvalues appear in \pm pairs, ensuring the algebraic multiplicity of each pair is even. In odd dimensions (n odd), the odd degree of p(\lambda) requires zero to be an eigenvalue with odd multiplicity, implying A is singular.^[32] Over the complex numbers, A is diagonalizable, as iA is Hermitian and thus unitarily diagonalizable, yielding eigenvalues of the form i \theta and -i \theta for real \theta. The Jordan canonical form contains no non-trivial Jordan blocks for non-zero eigenvalues, due to this diagonalizability. The geometric multiplicity matches the algebraic multiplicity for all eigenvalues.^[31] In the real canonical form, there exists an orthogonal matrix Q such that Q^T A Q is block diagonal, consisting of 2×2 rotation-scaled blocks

\begin{pmatrix} 0 & \lambda_k \\ -\lambda_k & 0 \end{pmatrix}, \quad \lambda_k > 0,

along with 1×1 zero blocks corresponding to the kernel. The dimension of the kernel (nullity, or corank, of A) has the same parity as n; for a non-singular A (which requires n even), this dimension is zero. This form underscores the pairing of non-zero eigenvalues.^[33]

Geometric Applications

Cross Product Representation

In three-dimensional Euclidean space, skew-symmetric matrices provide a matrix representation of the cross product operation. For a vector \mathbf{v} = (v_1, v_2, v_3)^\top \in \mathbb{R}^3, the associated skew-symmetric matrix K_{\mathbf{v}} is defined as

K_{\mathbf{v}} = \begin{pmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{pmatrix}.

This matrix satisfies K_{\mathbf{v}} \mathbf{w} = \mathbf{v} \times \mathbf{w} for any \mathbf{w} \in \mathbb{R}^3, where \times denotes the standard cross product.^[34] The mapping \mathbf{v} \mapsto K_{\mathbf{v}} is linear over \mathbb{R}, preserving addition and scalar multiplication: K_{\mathbf{u} + \mathbf{v}} = K_{\mathbf{u}} + K_{\mathbf{v}} and K_{c \mathbf{v}} = c K_{\mathbf{v}} for \mathbf{u}, \mathbf{v} \in \mathbb{R}^3 and c \in \mathbb{R}. Moreover, this mapping extends to the Lie bracket structure, establishing an isomorphism of Lie algebras between \mathfrak{so}(3), the space of $3 \times 3 skew-symmetric matrices under the commutator bracket, and \mathbb{R}^3 under the cross product.^[34] A useful norm relation for this representation is the Frobenius norm \|K_{\mathbf{v}}\|_F = \sqrt{2} \|\mathbf{v}\|_2, where \|\cdot\|_2 is the Euclidean norm on \mathbb{R}^3; this follows from squaring the off-diagonal entries of K_{\mathbf{v}} and summing under the Frobenius norm definition \|A\|_F = \sqrt{\sum_{i,j} a_{ij}^2}. While analogous representations arise in higher dimensions through the adjoint action of \mathfrak{so}(n) on \mathbb{R}^n, the direct equivalence to a cross product is unique to three dimensions.^[34]

Infinitesimal Rotations

Skew-symmetric matrices form the Lie algebra \mathfrak{so}(n) of the special orthogonal group \mathrm{SO}(n), consisting of all n \times n real skew-symmetric matrices under the Lie bracket [A, B] = AB - BA. This bracket captures the infinitesimal composition of rotations, reflecting the non-commutative nature of the orthogonal group. The exponential map \exp: \mathfrak{so}(n) \to \mathrm{SO}(n), defined by the matrix exponential \exp(A) = \sum_{k=0}^\infty \frac{A^k}{k!}, associates each skew-symmetric matrix A with a rotation matrix, providing a local parametrization of the group near the identity.^[35]^[36]^[37] The basis elements of \mathfrak{so}(n) serve as infinitesimal generators of rotations in specific planes. For each pair of distinct indices i < j, the matrix E_{ij} with a 1 in position (i,j), a -1 in position (j,i), and zeros elsewhere generates an infinitesimal rotation in the i-j plane. These n(n-1)/2 basis matrices span \mathfrak{so}(n), and any general element is a linear combination \sum_{i<j} \theta_{ij} E_{ij}, corresponding to simultaneous infinitesimal rotations by angles \theta_{ij} in each plane. The dimension n(n-1)/2 matches the number of independent rotation parameters in \mathrm{SO}(n).^[38]^[39] In physics, particularly quantum mechanics, angular momentum operators play the role of generators for rotations but are Hermitian to ensure real eigenvalues for observables. The infinitesimal rotation generators are then skew-Hermitian, related to these operators by a factor of i (up to constants like \hbar), linking the real skew-symmetric structure of \mathfrak{so}(3) to the complex skew-Hermitian Lie algebra \mathfrak{su}(2), which is isomorphic. This connection underlies the representation of rotations in quantum systems, where the commutation relations [J_x, J_y] = i \hbar J_z (and cyclic permutations) mirror the Lie algebra structure.^[40]^[41] Noether's theorem establishes that rotational invariance of the Lagrangian in physical systems implies the conservation of angular momentum, with skew-symmetric matrices encoding the associated symmetries. For a system invariant under continuous rotations, the theorem yields a conserved current corresponding to the angular momentum tensor, whose components are generated by elements of \mathfrak{so}(n). This principle explains the persistence of angular momentum in isolated systems, from classical mechanics to quantum field theory.^[42]^[43]

Coordinate-Free Perspective

In the coordinate-free framework, a skew-symmetric matrix arises as the matrix representation of an alternating bilinear form on a finite-dimensional vector space V over a field F of characteristic not equal to 2. An alternating bilinear form \omega: V \times V \to F is a bilinear map satisfying \omega(v, v) = 0 for all v \in V, which equivalently implies \omega(u, v) = -\omega(v, u) for all u, v \in V.^[21] The space of all such forms, denoted \mathrm{Alt}^2(V; F), is naturally isomorphic to the second exterior power \Lambda^2 V^* of the dual space V^*, where elements of \Lambda^2 V^* are skew-symmetric contravariant 2-tensors, or bivectors in the dual sense.^[44] Given a basis \{e_1, \dots, e_n\} for V, the associated dual basis \{e^1, \dots, e^n\} for V^* yields the coordinate matrix A = (a_{ij}) of \omega, defined by a_{ij} = \omega(e_i, e_j). This matrix satisfies A^T = -A, confirming its skew-symmetry, and the form can be expressed in the exterior algebra as

\omega = \sum_{i < j} a_{ij} \, e^i \wedge e^j.

This identification highlights that skew-symmetric matrices parametrize the components of bivectors in \Lambda^2 V^*, emphasizing the intrinsic tensorial nature independent of any specific inner product.^[21]^[44] Under a change of basis specified by an invertible matrix P \in \mathrm{GL}(n, F), where the new basis vectors are the columns of P applied to the old basis, the matrix representation transforms via the congruence A' = P^T A P. This law preserves skew-symmetry, as (P^T A P)^T = P^T A^T P = -P^T A P, ensuring the abstract form \omega remains unchanged while its coordinates adapt to the new frame.^[45] Certain quantities are basis-independent: the trace of A is always zero, reflecting \omega(e_i, e_i) = 0 for all basis vectors, and the determinant transforms as \det(A') = \det(P)^2 \det(A), rendering it invariant up to positive scaling in contexts where basis changes preserve volume up to sign, such as orthogonal transformations.^[21] This perspective connects directly to the study of alternating forms, where \omega defines pairings on V.^[45]

Connections to Forms

Alternating Bilinear Forms

A skew-symmetric matrix A \in M_n(\mathbb{R}) (or more generally over a field of characteristic not equal to 2) defines an alternating bilinear form \omega: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} by \omega(\mathbf{u}, \mathbf{v}) = \mathbf{u}^T A \mathbf{v}. This form is bilinear by the properties of matrix multiplication and alternating because \omega(\mathbf{v}, \mathbf{v}) = \mathbf{v}^T A \mathbf{v} = 0 (since A^T = -A implies the quadratic form vanishes) and \omega(\mathbf{u}, \mathbf{v}) = -\omega(\mathbf{v}, \mathbf{u}).^[21]^[46] The map sending a skew-symmetric matrix A to the alternating bilinear form \omega(\mathbf{u}, \mathbf{v}) = \mathbf{u}^T A \mathbf{v} is an isomorphism between the vector space of n \times n skew-symmetric matrices and the space of alternating bilinear forms on \mathbb{R}^n. Under this identification, the matrix A is the coordinate matrix of \omega with respect to the standard basis.^[21]^[47] The form \omega is non-degenerate—meaning \omega(\mathbf{u}, \mathbf{v}) = 0 for all \mathbf{v} implies \mathbf{u} = \mathbf{0}—if and only if the matrix A is invertible. Equivalently, the radical of \omega (the set of \mathbf{u} such that \omega(\mathbf{u}, \mathbf{v}) = 0 for all \mathbf{v}) is trivial precisely when \ker A = \{\mathbf{0}\}.^[21]^[48] In even dimensions n = 2m, a non-degenerate alternating bilinear form is known as a symplectic form, and linear transformations preserving such a form maintain the volume up to sign, with the determinant being 1 for the standard symplectic group. The Pfaffian of A, satisfying \operatorname{pf}(A)^2 = \det A, relates to the oriented volume induced by the form.^[47]^[21] Any alternating bilinear form admits a canonical form under basis change: if the rank is $2k (always even), there is a basis in which the matrix is block diagonal with k copies of the $2 \times 2 block \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} along the diagonal and zeros elsewhere. For non-degenerate cases in even dimensions, the Darboux theorem guarantees a basis (called a symplectic or Darboux basis) where the matrix takes the standard block form consisting of m such $2 \times 2 blocks.^[48]^[21]

Relation to Orthogonal and Symplectic Groups

Skew-symmetric matrices constitute the Lie algebra \mathfrak{so}(n, \mathbb{R}) of the orthogonal group O(n, \mathbb{R}), which comprises all n \times n real matrices g satisfying g^\top g = I_n, thereby preserving the standard symmetric bilinear form \langle u, v \rangle = u^\top v. This preservation condition implies that for a one-parameter subgroup g(t) with g(0) = I_n and g'(0) = A, differentiation yields A^\top + A = 0, confirming that elements of the Lie algebra are precisely the skew-symmetric matrices.^[49] The exponential map \exp: \mathfrak{so}(n, \mathbb{R}) \to SO(n, \mathbb{R}) associates each skew-symmetric matrix A with a rotation matrix \exp(A), which preserves the associated quadratic form q(x) = x^\top x.^[50] In even dimensions $2n, skew-symmetric matrices also play a central role in defining alternating bilinear forms preserved by the symplectic group Sp(2n, \mathbb{R}), the subgroup of GL(2n, \mathbb{R}) consisting of matrices g such that g^\top J g = J, where J is the standard symplectic matrix

J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix},

a block-diagonal skew-symmetric matrix representing the non-degenerate alternating form \omega(u, v) = u^\top J v. The Lie algebra \mathfrak{sp}(2n, \mathbb{R}) consists of all $2n \times 2n matrices X satisfying X^\top J + J X = 0, derived from the infinitesimal preservation condition.^[51] Although elements of \mathfrak{sp}(2n, \mathbb{R}) are not skew-symmetric in general, the fixed matrix J underscores how skew-symmetric matrices parameterize such alternating forms.^[51] The orthogonal and symplectic groups thus arise as preservers of distinct classes of bilinear forms: symmetric forms for the former (with Lie algebra skew-symmetric relative to the identity) and alternating forms for the latter (with Lie algebra satisfying the twisted skew-symmetry X^\top J + J X = 0). For a general invertible skew-symmetric matrix A, the group \{ g \in GL(2n, \mathbb{R}) \mid g^\top A g = A \} is isomorphic to Sp(2n, \mathbb{R}) via change of basis, preserving the associated non-degenerate alternating form \omega(u, v) = u^\top A v; if A is singular, this group enlarges accordingly, corresponding to a degenerate case.^[51]

Generalizations

Skew-Symmetrizable Matrices

A skew-symmetrizable matrix is a square real matrix A for which there exists a diagonal matrix D with positive diagonal entries such that D A D^{-1} is skew-symmetric. This condition implies that A is similar to a skew-symmetric matrix via a diagonal similarity transformation, generalizing the class of skew-symmetric matrices (which correspond to the special case D = I).^[52] The eigenvalues of a skew-symmetrizable matrix are either zero or purely imaginary, occurring in complex conjugate pairs, mirroring the spectral properties of skew-symmetric matrices due to the similarity. These matrices are also linked to specific sign patterns in combinatorial matrix theory, where the existence of such a D requires compatible signs in off-diagonal entries to allow balancing via positive scalings.^[53] In applications, skew-symmetrizable matrices arise in stability analysis of dynamical systems, such as epidemic models, where the Jacobian matrix satisfying a weighted skew-symmetry condition ensures global asymptotic stability under certain positivity assumptions. They also connect to generalized inverses, where the structure facilitates computations analogous to those for skew-symmetric cases, and to totally nonnegative matrices through combinatorial frameworks like cluster algebras, where skew-symmetrizable exchange matrices underpin classifications of finite-type structures.^[54]

Matrices over Complex Numbers

Over the complex numbers, a skew-symmetric matrix is defined as a square matrix A \in M_n(\mathbb{C}) satisfying A^T = -A, where ^T denotes the transpose without conjugation.^[55] This definition parallels the real case but lacks the additional structure imposed by reality of entries, leading to distinct spectral behavior. Unlike real skew-symmetric matrices, whose eigenvalues are purely imaginary or zero, complex skew-symmetric matrices can have eigenvalues with nonzero real parts.^[56] The characteristic polynomial p_A(\lambda) = \det(\lambda I - A) satisfies p_A(\lambda) = p_A(-\lambda), implying that nonzero eigenvalues occur in pairs \lambda and -\lambda, where \lambda \in \mathbb{C} need not lie on the imaginary axis.^[57] For instance, consider the $2 \times 2 matrix

A = \begin{pmatrix} 0 & 1+i \\ -(1+i) & 0 \end{pmatrix},

which is skew-symmetric since A^T = -A. Its characteristic polynomial is \lambda^2 + (1+i)^2 = \lambda^2 + (1 + 2i - 1) = \lambda^2 + 2i, with roots \lambda = \pm \sqrt{-2i} = \pm (1 - i), eigenvalues having real part 1 and -1, respectively. Complex skew-symmetric matrices are not necessarily diagonalizable over \mathbb{C}; Jordan blocks can appear, particularly for the eigenvalue 0, though blocks for paired eigenvalues \lambda and -\lambda must have matching structures. The possible Jordan canonical forms are constrained by the skew-symmetry, often involving blocks of even size for nonzero eigenvalues.^[56]^[57] A key distinction arises in comparison to skew-Hermitian matrices, defined by A^* = -A where ^* is the conjugate transpose. Skew-Hermitian matrices serve as the direct complex generalization of real skew-symmetric matrices, inheriting properties like purely imaginary eigenvalues and unitarily diagonalizable forms (up to the factor of i). Specifically, if K is skew-Hermitian, then iK is Hermitian, with real eigenvalues. In contrast, complex skew-symmetric matrices do not generally satisfy A^* = -A unless their entries are purely imaginary, and thus lack these Hermitian-like spectral guarantees.^[58] In applications, complex skew-symmetric matrices play a role in the structure of complex Lie algebras, particularly as the defining elements of \mathfrak{so}(n, \mathbb{C}), the Lie algebra of the complex orthogonal group O(n, \mathbb{C}), consisting of all n \times n skew-symmetric matrices over \mathbb{C}. This algebra is semisimple for n \geq 3 and intersects with \mathfrak{sl}(n, \mathbb{C}) (trace-zero matrices) in a subspace of codimension 1 when n is odd. However, due to the prevalence of Hermitian structures in quantum mechanics and other complex analysis contexts, skew-symmetric matrices over \mathbb{C} receive less attention in the literature compared to their real or skew-Hermitian counterparts.^[59]

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[PDF] Quadratic Polynomials - Penn Math
A matrix C is called skew-symmetric if CT = −C . (a) Give an example of a skew-symmetric 2 × 2 matrix (other than the 0 matrix). (b) Show that C is skew- ...
[19]
Math21b, Fall 2003, Linear Algebra and Differential Equations
For (1x1) matrices, this is the usual exponential function. Verify that if A is a skew symmetric matrix, then eA is an orthogonal matrix. Hint. Check first ...
[20]
[PDF] Cross-Products and Rotations
Nov 23, 2003 · k/k! . Now, every 3-by-3 skew-symmetric matrix S determines a vector v such that S = v. ¢. ; then. Sv = o , S. 2. = vv. T. – ||v||. 2. ·I , S. 3.
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[PDF] BILINEAR FORMS The geometry of Rn is controlled algebraically by ...
Among the three types of bilinear forms we have defined (symmetric, skew-symmetric, alternating), the first and third types are more basic than the second.
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[PDF] Exponentials and Rotations - UMD MATH
Jul 21, 2021 · If A is a skew-symmetric n × n matrix, then eA is orthog- onal. Additionally, det(eA) = 1, so eA is a rotation matrix. So what does a skew- ...
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[PDF] Vector spaces - Purdue Math
Aug 23, 2024 · Dimension of the space of skew-symmetric matrices of size n × n is n(n −. 1)/2. The main use of a basis is the possibility to represent ...
[24]
[PDF] Chapter 10 Euclidean Spaces - UPenn CIS
define the inner product of two matrices A, B 2 Mn(R) as. hA, Bi = n. X i,j=1 ... The (real) vector space of n⇥n skew symmetric matrices is denoted by so(n).
[25]
[PDF] Midterm, October 18, 2011, with solutions Linear Algebra I Cross out ...
The dimension of the space equals the number of elements above the diagonal, i.e., equals n(n − 1)/2. A basis elements can be chosen to vanish except for ...
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CS 6210: Matrix Computations
The two spaces are orthogonal to each other in the Frobenius inner product; and for any matrix A ∈ R n × n , there is a unique decomposition into a symmetric ...
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A note on skew-symmetric determinants - Cambridge University Press
There are two classical results about a skew-symmetric matrix A: (I) When n is odd, then deM = 0. (II) When n is even, then det A = (pn(A))2, where pn(A) is a ...
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Pfaffian | Wolfram Function Repository
The Pfaffian of a skew-symmetric matrix m is an integer-coefficient polynomial in the entries of m whose square is the determinant of m.
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https://www.tandfonline.com/doi/full/10.1080/0025570X.2024.2378680
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Graph theory and Pfaffian representations of Ising partition function
Dec 27, 2013 · The partition function of an Ising model on an arbitrary planar graph can be represented as the Pfaffian of a skew-symmetric matrix associated to the graph.
[31]
[PDF] Chapter 6 Eigenvalues and Eigenvectors
A symmetric matrix (ST = S) can be compared to a real number. A skew-symmetric matrix (AT = −A) can be compared to an imaginary number. An orthogonal matrix. ( ...
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Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely ...
We prove that eigenvalues of a real skew-symmetric matrix are zero or purely imaginary and the rank of the matrix is even. We use the diagonalization of ...
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[PDF] Spectral Theorems in Euclidean and Hermitian Spaces - CIS UPenn
Spectral Theorems. Theorem 11.4.4 For every skew symmetric matrix A there is an orthog- onal matrix P and a block diagonal matrix D such that A = PDP. , where.Missing: theory | Show results with:theory
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[PDF] A QUICK NOTE ON ORTHOGONAL LIE ALGEBRAS
(R3, ×) is a Lie algebra, which is isomorphic to so(3,R). Proof: We have already mentioned that × is skew-symmetric. Moreover, the cyclic sum over x, y and ...
[35]
[PDF] Lie Groups and Mechanics: an introduction - arXiv
The Lie algebra so(n) of the rotation group SO(n) consists of skew-symmetric n × n matrices. 0. 2.2. Adjoint representation of G. The composition Ig = Rg−1 ...
[36]
[PDF] Lecture Notes in Lie Groups - arXiv
Apr 7, 2011 · so(n, R) skew– symmetric square real matrices, with. Lie bracket the ... unitary matrices; the Lie algebra will then map to skew–Hermitian ...
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[PDF] arXiv:2010.05020v1 [math.OC] 10 Oct 2020
Oct 10, 2020 · Any matrix Q ∈ SO(N) can be represented as a matrix ex- ponential function of a skew-symmetric matrix. The converse is also true: the.
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[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
... Lie algebra with the commutator given by the cross-product, is isomorphic (as a Lie algebra) to so(3, R). Page 37. Exercises. 37. (2) Let ϕ: so(3, R) → R3 be ...
[39]
[PDF] Ensemble Control on Lie Groups - arXiv
Aug 7, 2020 · Basics of the Lie algebra so(n). The Lie algebra so(n) is the vector space containing all n × n real skew-symmetric matrices, which has ...
[40]
[PDF] Lectures 6 and 7 SO(3) and Conserved Quantities
The generators of the group are the angular momentum operators. • Note that these operators obey the same commutation relation as the three generators did for.
[41]
[PDF] Lecture 14 Angular momentum operator algebra
Consider the three Hermitian angular momentum operators. ˆ. Jx,. ˆ. Jy and ˆJz ... commutation relations (14.1) represent the Lie algebra of the SO(3) group.Missing: skew- | Show results with:skew-
[42]
[PDF] Noether's Theorem - Physics Courses
This is the conserved angular momentum about the z axis. If instead the particle moved in a potential U(y), independent of x, then writing. L = 1.
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[PDF] Conservation Laws
◇ translation invariance leads to linear momentum conservation. ◇ rotational invariance leads to angular momentum conservation. □ Familiar conserved ...<|control11|><|separator|>
[44]
[PDF] 4 Exterior algebra - People
We can add alternating forms and multiply by scalars so they form a vector space, isomorphic to the space of skew-symmetric n × n matrices. ... 2-vector of ...
[45]
[PDF] Bilinear forms and their matrices
Mar 11, 2011 · A bilinear form H is called symmetric if H(v, w) = H(w, v) for all v, w ∈ V . A bilinear form H is called skew-symmetric if H(v, w) = −H(w, v) ...
[46]
[PDF] Skew-symmetric bilinear forms - Fachbereich Mathematik und Statistik
Dec 9, 2020 · Let f be a skew-symmetric bilinear form f is alternating if and only if char(K) 6= 2. Proof: Let f be any skew-symmetric bilinear form over a ...Missing: coordinate- free
[47]
[PDF] bilinear forms
If <-, -> is alternating, then we know. 0 = <v + w,v + w> = <v,v> + <v, w> + <w,v> + <w, w> = <v, w> + <w,v>. Thus <-, -> is skew-symmetric. If the ...
[48]
[PDF] 1. Linear algebra preliminaries 1.1. Some facts about bilinear forms ...
Skew-symmetric bilinear forms. Proposition. Let ψ be a skew-symmetric bilinear form. Then, there exists a basis with respect to which ψ is given by the matrix.
[49]
Appendix D The Orthogonal Group and Its Lie Algebra
identifies the skew-symmetric linear maps as the Lie algebra o(d) of O(d) ... The Lie bracket for o(d) is the usual bracket of matrices [ξ1,ξ2] = ξ1ξ2 − ξ2ξ1.<|control11|><|separator|>
[50]
[PDF] The Exponential Map, Lie Groups, and Lie Algebras - UPenn CIS
The Lie algebra so(n,R) consisting of real skew symmet- ric n×n matrices is the corresponding set of infinitesimal rotations. The geometric link between a Lie ...
[51]
[PDF] 18.745 F20 Lecture 06: Classical Lie Groups - MIT OpenCourseWare
An important role in the theory of Lie groups is played by the algebra of quaternions, which is the only noncommu- tative finite dimensional division algebra ...<|control11|><|separator|>
[52]
Least Squares Symmetrizable Solutions for a Class of Matrix ...
Definition 1. A real matrix A is called a symmetrizable (skew-symmetrizable) matrix if A is similar to a symmetric (skew-symmetric) matrix A.
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Mutation classes of skew-symmetrizable 3×3 matrices
Sep 27, 2012 · Let us recall that an integer matrix B is skew-symmetrizable if DB is skew-symmetric for some diagonal matrix. D with positive diagonal entries.
[54]
[PDF] cluster algebras of finite type and positive symmetrizable matrices
The mutation-equivalence class of a skew-symmetrizable matrix B satisfies the equivalent conditions in Theorem 1.1 if and only if B satisfies: (4) Every ...
[55]
[PDF] Canonical forms for complex matrix congruence and ... - arXiv
Sep 16, 2007 · Any square complex matrix A can be represented uniquely as A = S +C, in which S is symmetric and C is skew-symmetric; it can also be represented.
[56]
The Jordan Canonical Forms of complex orthogonal and skew ...
We study the Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices, and consider some related results.
[57]
Canonical form of complex skew symmetric matrix - MathOverflow
Jul 9, 2021 · ... Horn and Johnson's matrix analysis. So do there exists canonical form of unitary similarity transformation for complex skew symmetric matrix?
[58]
Hermitian and Skew-Hermitian Matrix - GeeksforGeeks
Aug 19, 2025 · A complex square matrix is said to be a skew-Hermitian matrix if the conjugate transpose matrix is equal to the negative of the original matrix.
[59]
[PDF] Classification of Semisimple Lie Algebras
For completeness, I will list the remaining classical algebras. The. Lie algebra so(n) consists of skew-symmetric matrices. Cartan's classification of so(2n ...