Permute definition math9/28/2023 ![]() Here is an example with the MATLAB lu function that illustrates how permuting a matrix can be done using the vector permutation representation. MATLAB functions that return permutation matrices can also return the permutation in vector form. In coding, memory can be saved by representing a permutation matrix as an integer vector, where is the column index of the within the th row of. A matrix is doubly stochastic if and only if it is a convex combination of permutation matrices. A classic result characterizes doubly stochastic matrices in terms of permutation matrices. Every reducible permutation matrix is permutation similar to a direct sum of irreducible permutation matrices.Īnother notable permutation matrix is the vec-permutation matrix, which relates to, where is the Kronecker product.Ī permutation matrix is an example of a doubly stochastic matrix: a nonnegative matrix whose row and column sums are all equal to. The next result shows that a reducible permutation matrix can be expressed in terms of irreducible permutation matrices. 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, , …. 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.Īn 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.Įxamples 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.
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