An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix that satisfies the identity
(1)
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where is the matrix transpose. For example,
(2)
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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
(3)
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Letting , the requirement becomes
(4)
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so an antisymmetric matrix must have zeros on its diagonal. The general antisymmetric matrix is of the form
(5)
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Applying to both sides of the antisymmetry condition gives
(6)
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Any square matrix can be expressed as the sum of symmetric and antisymmetric parts. Write
(7)
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But
(8)
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(9)
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so
(10)
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which is symmetric, and
(11)
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which is antisymmetric.
All antisymmetric matrices of odd dimension are singular. This follows from the fact that
(12)
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So, by the properties of determinants,
(13)
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(14)
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Therefore, if is odd, then
(15)
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thus proving all antisymmetric matrices of odd dimension are singular.
The set of antisymmetric matrices is denoted . is a vector space, and the commutator
(16)
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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 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 must be an antisymmetric matrix.
The matrix exponential map of an antisymmetric matrix is an orthogonal matrix.