Skew-symmetric matrix

This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices

Definition

Verbal definition

A square matrix is termed a skew-symmetric matrix if is negative equals its matrix transpose. In symbols, a matrix $A$ is termed skew-symmetric if $-A=A^{T}$.

Algebraic definition

Suppose $n$ is a positive integer and $A=(a_{ij})_{1\leq i\leq n,1\leq j\leq n}$ is a $n\times n$ matrix. We say that $A$ is skew=symmetric if the following holds:

$a_{ij}=-a_{ji}\ \forall \ i,j\in \{1,2,\dots ,n\}$

Because the equality condition is symmetric in $i,j$, it suffices to check it for $i<j$, so the above definition if equivalent to:

$a_{ij}=a_{ji}\ \forall \ i\leq j\in \{1,2,\dots ,n\}$

Note that it is important to include the condition on the diagonal elements, and that this condition forces the diagonal elements to all equal zero.