One consequence of Theorem 1 is that for any irreducible permutation matrix. There exists a permutation matrix such that.For a permutation matrix the following conditions are equivalent. Where and are square, nonempty submatrices. Recall that a matrix is irreducible if there does not exist a permutation matrix such that The shift matrix plays a fundamental role in characterizing irreducible permutation matrices. The following animated gif superposes MATLAB spy plots of, , вЂ¦. The matrix has two diagonals of s, which move up through the matrix as it is powered: for and. It is easy to show that, which means that the eigenvalues of are, where is the th root of unity. A general permutation matrix can be written as a product of elementary permutation matrices, where is such that. Such a matrix is symmetric and so satisfies, and it has determinant. It can be written, where is the th column of. Note that is a symmetric Hankel matrix and is a circulant matrix.ДЄn elementary permutation matrix differs from in just two rows and columns, and, say. Pre- or postmultiplying a matrix by shifts the rows or columns, respectively, one place forward and moves the first one to the last position-that is, it cyclically permutes the rows or columns. Pre- or postmultiplying a matrix by reverses the order of the rows and columns, respectively. A permutation matrix that has the desired reordering effect is constructed by doing the same operations on the identity matrix.Д®xamples of permutation matrices are the identity matrix, the reverse identity matrix, and the shift matrix (also called the cyclic permutation matrix), illustrated for by Premultiplying a matrix by reorders the rows and postmultiplying by reorders the columns. The total number of permutation matrices is. Such a matrix, say, is orthogonal, that is,, so it is nonsingular and has determinant. A permutation matrix is a square matrix in which every row and every column contains a single and all the other elements are zero.

Permute rows of a matrix