Permutation matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly in <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix theory</a>, a permutation matrix is a square <a href="/facts/Binary_matrix/pS4BnBzj">binary matrix</a> that has exactly one entry of 1 in each row and each column with all other entries 0.: 26  An n × n permutation matrix can represent a <a href="/facts/Permutation/JSw9ukmv">permutation</a> of n elements. Pre-<a href="/facts/Matrix_multiplication/lYDB6Ro2">multiplying</a> an n-row matrix M by a permutation matrix P, forming PM, results in permuting the rows of M, while post-multiplying an n-column matrix M, forming MP, permutes the columns of M.
Every permutation matrix P is <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal</a>, with its <a href="/facts/Invertible_matrix/fPqXk3V8">inverse</a> equal to its <a href="/facts/Transpose/8wmsagGS">transpose</a>: 
 
 
 
 
 P
 
 −
 1
 
 
 =
 
 P
 
 
 T
 
 
 
 
 
 {\displaystyle P^{-1}=P^{\mathsf {T}}}
 
.: 26  Indeed, permutation matrices can be <a href="/facts/Characterization_(mathematics)/mmT1knFI">characterized</a> as the orthogonal matrices whose entries are all <a href="/facts/Non-negative/z0xollg1">non-negative</a>.

Permutation matrix open-in-new

Permutation matrix