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Antisymmetric Matrix -- from Wolfram MathWorld

02 Apr 20252 min read

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Antisymmetric Matrix — from Wolfram MathWorld

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An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix that satisfies the identity A=-A^(T) (1) where A^(T) is the matrix transpose. For example, A=[0 -1; 1 0] (2) is antisymmetric. A matrix m may be tested to see if it is antisymmetric in the Wolfram Language using AntisymmetricMatrixQ[m]. In component notation, this becomes a_(ij)=-a_(ji). (3) Letting k=i=j, the requirement becomes a_(kk)=-a_(kk), (4) so an antisymmetric matrix must…

An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix that satisfies the identity

(1)

where A^(T) is the matrix transpose. For example,

(2)

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)

Letting k=i=j , the requirement becomes

(4)

so an antisymmetric matrix must have zeros on its diagonal. The general 3×3 antisymmetric matrix is of the form

[0 a_(12) a_(13); -a_(12) 0 a_(23); -a_(13) -a_(23) 0].

(5)

Applying A^(-1) to both sides of the antisymmetry condition gives

(6)

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

(7)

But

A=[a_(11) a_(12) ... a_(1n); a_(21) a_(22) ... a_(2n); | | ... |; a_(n1) a_(n2) ... a_(nn)]

(8)

A^(T)=[a_(11) a_(21) ... a_(n1); a_(12) a_(22) ... a_(n2); | | ... |; a_(1n) a_(2n) ... a_(nn)],

(9)

so

A+A^(T)=[2a_(11) a_(12)+a_(21) ... a_(1n)+a_(n1); a_(12)+a_(21) 2a_(22) ... a_(2n)+a_(n2); | | ... |; a_(1n)+a_(n1) a_(2n)+a_(n2) ... 2a_(nn)],

(10)

which is symmetric, and

A-A^(T)=[0 a_(12)-a_(21) ... a_(1n)-a_(n1); -(a_(12)-a_(21)) 0 ... a_(2n)-a_(n2); | | ... |; -(a_(1n)-a_(n1)) -(a_(2n)-a_(n2)) ... 0],

(11)

which is antisymmetric.

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

(12)

So, by the properties of determinants,

Therefore, if is odd, then

(15)

thus proving all antisymmetric matrices of odd dimension are singular.

The set of n×n antisymmetric matrices is denoted o(n) . o(n) is a vector space, and the commutator

(16)

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 A(t) is a path of orthogonal matrices through A(0)=I , i.e., A(t)A^(T)(t)=I for all . The derivative at t=0 of both sides must be equal so dA/dt(0)+dA^(T)/dt(0)=0 . That is, the derivative of A(t) at the identity must be an antisymmetric matrix.