Definition:Permutation Matrix
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Definition
A permutation matrix (of order $n$) is an $n \times n$ square matrix with:
Examples
$3 \times 3$ Permutation Matrix
Definition:Permutation Matrix/Examples/3 x 3
Full Rook Matrix
An $8 \times 8$ permutation matrix is known as a full rook matrix.
For example:
- $\mathbf A = \begin{bmatrix}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}$
That is, it is a rook matrix in which each row and column has a $1$ in it.
Also see
- Results about permutation matrices can be found here.
Sources
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.3$: $m \times n$ matrices
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): permutation matrix