Antisymmetric Matrix

An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix whose entries are real numbers or complex numbers and which satisfies the identity

where is the matrix transpose. For example,

is antisymmetric.

A matrix may be tested to see if it is antisymmetric in the Wolfram Language using AntisymmetricMatrixQ[m].

In component notation, this becomes

Letting , the requirement becomes

so an antisymmetric matrix must have zeros on its diagonal. The general antisymmetric matrix is of the form

If is nonsingular, applying its matrix inverse to both sides of the antisymmetry condition gives

Any square matrix can be expressed as the sum of symmetric and antisymmetric parts. Write

But

so

which is symmetric, and

which is antisymmetric.

All antisymmetric matrices of odd dimension are singular. This follows from the fact that

So, by the properties of determinants,

Therefore, if is odd, then

thus proving all antisymmetric matrices of odd dimension are singular.

The set of real antisymmetric matrices is denoted . is a real vector space, and the commutator

of two antisymmetric matrices is antisymmetric. Hence, the antisymmetric matrices are a Lie algebra, which is related to the Lie group of orthogonal matrices. In particular, suppose is a differentiable path of orthogonal matrices through , i.e., for all . The derivative at of both sides must be equal so . That is, the derivative of at the identity matrix must be an antisymmetric matrix.

The matrix exponential of a real antisymmetric matrix is an orthogonal matrix.