Linear and multilinear algebra; matrix theory Definition

Skew-symmetric matrix

Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ square matrix over $R$ . The matrix $A$ is skew-symmetric when $A^T = -A$ , where $A^T$ is the transpose of $A$ . Equivalently, $a_{ji} = -a_{ij}$ for all $i$ and $j$ .

Remarks

If $2$ is invertible in $R$ , for example over $\R$ , then each diagonal entry of a skew-symmetric matrix is $0$ . Over a field of characteristic not $2$ , each square matrix is the sum of a symmetric matrix and a skew-symmetric matrix: $A = \tfrac12 (A + A^T) + \tfrac12 (A - A^T)$ . A real skew-symmetric matrix of odd order has determinant $0$ . The matrix $\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$ is skew-symmetric.

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Matrix definition
Square matrix definition
Transpose definition
Skew-symmetric matrix definition

Tag	`03RNVFDEAN`
URI	`m2://03RNVFDEAN`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Skew-symmetric matrix
Uses	Transpose, Square matrix

Skew-symmetric matrix

Definition

Remarks

Reading path

Further reading

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