Skew-Symmetric Matrix

A Skew-Symmetric Matrix is a square matrix, [math]\displaystyle{ A=[a_{ij}]_{\substack{i=1\dots m\\j=1\dots n}} }[/math], whose matrix transpose is also its negative, that is when [math]\displaystyle{ a_{ij} = \begin{cases} -a_{ji} & \quad \text{if } i\#j\\ 0 & \quad \text{if } i=j\\ \end{cases} }[/math]

• AKA: Antisymmetric/Antimetric Matrix.
• Example(s):

  • [math]\displaystyle{ A=\begin{bmatrix}0 & 2 & 3 \\-2 & 0 & 5 \\-3 & -5 & 0 \end{bmatrix} }[/math] because [math]\displaystyle{ a_{11}=a_{22}=a_{33}=0 }[/math] and [math]\displaystyle{ a_{12}=-a_{21}, a_{13}=-a_{31}, a_{23}=-a_{32} }[/math]

• Counter-Example(s):
  • an Identity Matrix.
  • a Non-Square Matrix.
• See: Transpose, Linear Algebra.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/skew-symmetric_matrix Retrieved:2015-2-20.
  • In mathematics, and in particular linear algebra, a skew-symmetric' (or antisymmetric or antimetric ) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the condition If the entry in the and is a_ij, i.e. then the skew symmetric condition is For example, the following matrix is skew-symmetric: : [math]\displaystyle{ \begin{bmatrix} 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end{bmatrix}. }[/math]

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