Permutation matrix

In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0.[1]: 26  An n × n permutation matrix can represent a permutation of n elements. Pre-multiplying 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 orthogonal, with its inverse equal to its transpose: .[1]: 26  Indeed, permutation matrices can be characterized as the orthogonal matrices whose entries are all non-negative.[2]

  1. ^ a b Artin, Michael (1991). Algebra. Prentice Hall. pp. 24–26, 118, 259, 322. ISBN 0-13-004763-5. OCLC 24364036.
  2. ^ Zavlanos, Michael M.; Pappas, George J. (November 2008). "A dynamical systems approach to weighted graph matching". Automatica. 44 (11): 2817–2824. CiteSeerX 10.1.1.128.6870. doi:10.1016/j.automatica.2008.04.009. S2CID 834305. Retrieved 21 August 2022. Let denote the set of orthogonal matrices and denote the set of element-wise non-negative matrices. Then, , where is the set of permutation matrices.