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 a i j {\textstyle a_{ij}} denotes the entry in the i {\textstyle i} -th row and j {\textstyle j} -th column, then the skew-symmetric condition is equivalent to
In characteristic not equal to 2, diagonal elements of a skew-symmetric matrix are zeros because each element must be its own negative.
Example The matrix
A = [ 0 2 − 45 − 2 0 − 4 45 4 0 ] {\displaystyle A={\begin{bmatrix}0&2&-45\\-2&0&-4\\45&4&0\end{bmatrix}}}
is skew-symmetric because
A T = [ 0 − 2 45 2 0 4 − 45 − 4 0 ] = − A . {\displaystyle A^{\textsf {T}}={\begin{bmatrix}0&-2&45\\2&0&4\\-45&-4&0\end{bmatrix}}=-A.}
Properties Throughout, we assume that all matrix entries belong to a field F {\textstyle \mathbb {F} } 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. The eigenvalues of a real skew-symmetric matrix are pure imaginary. If A {\textstyle A} is a real skew-symmetric matrix, then I + A {\textstyle I+A} is invertible, where I {\textstyle I} is the identity matrix. If A {\textstyle A} is a skew-symmetric matrix then A 2 {\textstyle A^{2}} is a symmetric negative semi-definite matrix.
Vector space structure 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 n × n {\textstyle n\times n} skew-symmetric matrices has dimension 1 2 n ( n − 1 ) . {\textstyle {\frac {1}{2}}n(n-1).}
Let Mat n {\displaystyle {\mbox{Mat}}_{n}} denote the space of n × n {\textstyle n\times n} matrices. A skew-symmetric matrix is determined by 1 2 n ( n − 1 ) {\textstyle {\frac {1}{2}}n(n-1)} scalars (the number of entries above the main diagonal); a symmetric matrix is determined by 1 2 n ( n + 1 ) {\textstyle {\frac {1}{2}}n(n+1)} scalars (the number of entries on or above the main diagonal). Let Skew n {\textstyle {\mbox{Skew}}_{n}} denote the space of n × n {\textstyle n\times n} skew-symmetric matrices and Sym n {\textstyle {\mbox{Sym}}_{n}} denote the space of n × n {\textstyle n\times n} symmetric matrices. If A ∈ Mat n {\textstyle A\in {\mbox{Mat}}_{n}} then
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