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Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e., () = (). Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e., () = ().
Multiplying a matrix M by either or on either the left or the right will permute either the rows or columns of M by either π or π −1.The details are a bit tricky. To begin with, when we permute the entries of a vector (, …,) by some permutation π, we move the entry of the input vector into the () slot of the output vector.
In other words, vec(A) is the vector obtained by vectorizing A in column-major order. Similarly, vec(A T) is the vector obtaining by vectorizing A in row-major order. In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator [1]
For our matrix (), we may start by swapping rows to provide the desired conditions for the n-th column. For example, we might swap rows to perform partial pivoting, or we might do it to set the pivot element a n , n {\displaystyle a_{n,n}} on the main diagonal to a non-zero number so that we can complete the Gaussian elimination.
Pivoting might be thought of as swapping or sorting rows or columns in a matrix, and thus it can be represented as multiplication by permutation matrices. However, algorithms rarely move the matrix elements because this would cost too much time; instead, they just keep track of the permutations.
There are three types of elementary row operations: Swapping two rows, Multiplying a row by a nonzero number, Adding a multiple of one row to another row. Using these operations, a matrix can always be transformed into an upper triangular matrix (possibly bordered by rows or columns of zeros), and in fact one that is in row echelon form.
A permutation matrix is a (0, 1)-matrix, all of whose columns and rows each have exactly one nonzero element.. A Costas array is a special case of a permutation matrix.; An incidence matrix in combinatorics and finite geometry has ones to indicate incidence between points (or vertices) and lines of a geometry, blocks of a block design, or edges of a graph.
Swap: Swap two rows of a matrix. Scale: Multiply a row of a matrix by a nonzero constant. Pivot: Add a multiple of one row of a matrix to another row. Two matrices A and B are row equivalent if it is possible to transform A into B by a sequence of elementary row operations.