2012年9月20日木曜日

Permutation Matrix

A permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere.

Example
The permutation matrix Pπ corresponding to the permutation :
\pi=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 2 & 5 & 3 \end{pmatrix},
is
P_\pi 
= 
\begin{bmatrix}
\mathbf{e}_{\pi(1)} \\
\mathbf{e}_{\pi(2)} \\
\mathbf{e}_{\pi(3)} \\
\mathbf{e}_{\pi(4)} \\
\mathbf{e}_{\pi(5)} 
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{e}_{1} \\
\mathbf{e}_{4} \\
\mathbf{e}_{2} \\
\mathbf{e}_{5} \\
\mathbf{e}_{3} 
\end{bmatrix}
=
\begin{bmatrix} 
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 
\end{bmatrix}.

Properties
P_{\pi}P_{\pi}^{T} = I (orthogonal matrices)

http://en.wikipedia.org/wiki/Permutation_matrix


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