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- AT = −A
or in component form, if A = (aij):
- aij = − aji for
all i and j.
For example, the following matrix is skew-symmetric:
A matrix A is skew-symmetric if AT = −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 − AT)/2; the "symmetric component" of A is C = (A + AT)/2; the matrix A is the sum of its symmetric and skew-symmetric components.
If A is skew-symmetric and x is vector then xTAx = 0.
Let A be a n×n skew-symmetric matrix. The determinant of A satisfies
- det(A) = det(AT) = 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.
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 λk 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 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 2r × 2r skew-symmetric matrix can be written in the form A = R Σ RT where R is orthogonal and
for real λk. The eigenvalues of this matrix are ±iλk. In the odd-dimensional case Σ has an additional row and column of zeros.
- φ : V × V → K
- 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 xTAx.
- n(n − 1)/2.
- [A,B] = AB - BA
It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric.
The image of the exponential map of a Lie algebra always lies in the connected component of the Lie group that contains the identity element. In the case of the Lie group O(n), this connected component is the special orthogonal group SO(n), consisting of all orthogonal matrices with determinant 1. So R = exp(A) 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.
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