NIMA LABORATORY

n mathematics, and in particular linear algebra, a skew-symmetric (or antisymmetric or antimetric^[1]) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation A = −A^T. If the entry in the i ^th row and j ^th column is a_ij, i.e. A = (a_ij) then the symmetric condition becomes a_ij = −a_ji. For example, the following matrix is skew-symmetric:

Properties

We assume that the the underlying field is not of characteristic 2: that is, that 1 + 1 ≠ 0 where 1 denotes the multiplicative identity and 0 the additive identity of the given field. Otherwise, a skew-symmetric matrix is just the same thing as a symmetric matrix.

Sums and scalar multiples of skew-symmetric matrices are again skew-symmetric. Hence, the skew-symmetric matrices form a vector space. Its dimension is n(n−1)/2.

Let Mat_n denote the space of n × n matrices. A skew-symmetric matrix is determined by n(n − 1)/2 scalars (the number of entries above the main diagonal); a symmetric matrixis determined by n(n + 1)/2 scalars (the number of entries on or above the main diagonal). If Skew_n denotes the space of n × n skew-symmetric matrices and Sym_n denotes the space of n × n symmetric matrices and then since Mat_n = Skew_n + Sym_n and Skew_n ∩ Sym_n = {0}, i.e.

where ⊕ denotes the direct sum. Let A ∈ Mat_n then

Notice that ½(A − A^T) ∈ Skew_n and ½(A + A^T) ∈ Sym_n. This is true for every square matrix A with entries from any field whose characteristic is different from 2.

As to equivalent conditions, notice that the relation of skew-symmetricity, A=-A^T, holds for a matrix A if and only if one has x^TAy =-y^TAx for all vectors x and y. This is also equivalent to x^TAx=0 for all x (one implication being obvious, the other a plain consequence of (x+y)^TA(x+y)=0 for all x and y).

All main diagonal entries of a skew-symmetric matrix must be zero, so the trace is zero. If A = (a_ij) is skew-symmetric, a_ij = −a_ij; hence a_ii = 0.

[edit]Determinant

Let A be a n×n skew-symmetric matrix. The determinant of A satisfies

det(A) = det(A^T) = det(−A) = (−1)ⁿdet(A). Hence det(A) = 0 when n is odd.

In particular, if n is odd, and since the underlying field is not of characteristic 2, the determinant vanishes. This result is called Jacobi's theorem, after Carl Gustav Jacobi(Eves, 1980).

The even-dimensional case is more interesting. It turns out that the determinant of A for n even can be written as the square of a polynomial in the entries of A (Theorem byThomas Muir):

det(A) = Pf(A)².

This polynomial is called the Pfaffian of A and is denoted Pf(A). Thus the determinant of a real skew-symmetric matrix is always non-negative.

The number of terms s(n) in the expansion of a skew-symmetric matrix of order n has been considered already by Cayley, Sylvester, and Pfaff. Due to cancellations, this number is quite small as compared the number of terms of a generic matrix of order n, which is n!. The sequence s(n) (sequence A002370 in OEIS) is

1, 0, 2, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, …

and it is encoded in the exponential generating function

The latter yields to the asymptotics (for n even)

The number of positive and negative terms are approximatively a half of the total, although their difference takes larger and larger positive and negative values as n increases (sequence A167029 in OEIS).

[edit]Spectral theory

The eigenvalues of a skew-symmetric matrix always come in pairs ±λ (except in the odd-dimensional case where there is an additional unpaired 0 eigenvalue). For a real skew-symmetric matrix the nonzero eigenvalues are all pure imaginary and thus are of the form iλ₁, −iλ₁, iλ₂, −iλ₂, … where each of the λ_k are real.

Real skew-symmetric matrices are normal matrices (they commute with their adjoints) and are thus subject to the spectral theorem, which states that any real skew-symmetric matrix can be diagonalized by a unitary matrix. Since the eigenvalues of a real skew-symmetric matrix are complex it is not possible to diagonalize one by a real matrix. However, it is possible to bring every skew-symmetric matrix to a block diagonal form by an orthogonal transformation. Specifically, every 2n × 2n real skew-symmetric matrix can be written in the form A = Q Σ Q^T where Q is orthogonal and

for real λ_k. The nonzero eigenvalues of this matrix are ±iλ_k. In the odd-dimensional case Σ always has at least one row and column of zeros.