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Concept
Skew-symmetric matrix
Summary
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to
The matrix
is skew-symmetric because
Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2. That is, we assume that 1 + 1 ≠ 0, where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix.
The sum of two skew-symmetric matrices is skew-symmetric.
A scalar multiple of a skew-symmetric matrix is skew-symmetric.
The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero.
If is a real skew-symmetric matrix and is a real eigenvalue, then , i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
If is a real skew-symmetric matrix, then is invertible, where is the identity matrix.
If is a skew-symmetric matrix then is a symmetric negative semi-definite matrix.
As a result of the first two properties above, the set of all skew-symmetric matrices of a fixed size forms a vector space. The space of skew-symmetric matrices has dimension
Let denote the space of matrices. A skew-symmetric matrix is determined by scalars (the number of entries above the main diagonal); a symmetric matrix is determined by scalars (the number of entries on or above the main diagonal). Let denote the space of skew-symmetric matrices and denote the space of symmetric matrices. If then
Notice that and This is true for every square matrix with entries from any field whose characteristic is different from 2. Then, since and
where denotes the direct sum.
Denote by the standard inner product on The real matrix is skew-symmetric if and only if
This is also equivalent to for all (one implication being obvious, the other a plain consequence of for all and ).
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