Skew-symmetric matrix

In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation:

A^T = −A

or in component form, if A = (a_ij):

a_ij = − a_ji   for

all i and j.

For example, the following matrix is skew-symmetric:

<math>\begin{bmatrix}

0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end{bmatrix}<math>

Properties

A matrix A is skew-symmetric if A^T = −A

If matrices A and B are both skew-symmetric: (AB)^T = BA

The "skew-symmetric component of a matrix A is the matrix B = (A − A^T)/2; the "symmetric component" of A is C = (A + A^T)/2; the matrix A is the sum of its symmetric and skew-symmetric components.

If A is skew-symmetric and x is vector then x^TAx = 0.

All main diagonal entries of a skew-symmetric matrix have to be zero, and so the trace is zero.

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

det(A) = det(A^T) = det(−A) = (−1)ⁿdet(A).

In particular, if n is odd the determinant vanishes. 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:

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.

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 eigenvalues are all pure imaginary and thus are of the form iλ₁, −iλ₁, iλ₂, −iλ₂, … where each of the λ_k are real.

Skew-symmetric matrices fall into the category of normal matrices and are thus subject to the spectral theorem, which states that any real or complex skew-symmetric matrix can be diagonalized by a unitary matrix. Since the eigenvalues 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 2r × 2r skew-symmetric matrix can be written in the form A = R Σ R^T where R is orthogonal and

<math>\Sigma = \begin{bmatrix}

\begin{matrix}0 & \lambda_1\\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\ 0 & \begin{matrix}0 & \lambda_2\\ -\lambda_2 & 0\end{matrix} & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & \begin{matrix}0 & \lambda_r\\ -\lambda_r & 0\end{matrix} \end{bmatrix}<math> for real λ_k. The eigenvalues of this matrix are ±iλ_k. In the odd-dimensional case Σ has an additional row and column of zeros.

Alternating forms

An alternating form φ on a vector space V over a field K is defined (if K doesn't have characteristic 2) to be a bilinear form

φ : V × V → K

such that

φ(v,w) = −φ(w,v).

Such a φ will be represented by a skew-symmetric matrix, once a basis of V is chosen; and conversely an n×n skew-symmetric matrix A on Kⁿ gives rise to an alternating form x^TAx.

Infinitesimal rotations

The skew-symmetric n×n matrices form a vector space of dimension

n(n − 1)/2.

This is the tangent space to the orthogonal group O(n) at the identity matrix. In a sense, then, skew-symmetric matrices can be thought of as infinitesimal rotations.

Another way of saying this is that the space of skew-symmetric matrices forms the Lie algebra o(n) of the Lie group O(n). The Lie bracket on this space is given by the commutator:

<math>[A,B] = AB - BA<math>

It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric.

The matrix exponential of a skew-symmetric matrix A is then an orthogonal matrix R:

<math>R=\exp(A)=\sum_{n=0}^\infty \frac{A^n}{n!}.<math>

Since the image of the exponential map always lies in the connected component of O(n) (which is denoted SO(n)), R will have determinant +1. It turns out that every orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix.

Related topics

• symmetric matrix
• skew-Hermitian matrix
• symplectic matrix