Skew-symmetric matrix

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

$A{\text{ skew-symmetric}}\quad \iff \quad A^{\textsf {T}}=-A.$

In terms of the entries of the matrix, if ${\textstyle a_{ij}}$ denotes the entry in the ${\textstyle i}$ -th row and ${\textstyle j}$ -th column, then the skew-symmetric condition is equivalent to

$A{\text{ skew-symmetric}}\quad \iff \quad a_{ij}=-a_{ji}.$

In characteristic not equal to 2, diagonal elements of a skew-symmetric matrix are zeros because each element must be its own negative.