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Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
(PA is equal to A with its rows permuted in the same way that those of P are), though not all permutation matrices move every row. A square matrix is a permutation matrix iff P has exactly one 1 in each row and in each column. If P is a permutation matrix then P−1 is also a permutation matrix and P−1=PT (permutation matrices are orthogonal). The matrices Pi used for pivoting are permutation matrices that only switch two rows.
Solving linear systems
Published in Qingwen Hu, Concise Introduction to Linear Algebra, 2017
Example 2.5.13. Let P=0010100000010100 $ P = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right] $ . Then P is a permutation matrix and hence the inverse can be obtained by taking the transpose of P:
Numerical Linear Algebra
Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009
Azmy S. Ackleh, Edward James Allen, Ralph Baker Hearfott, Padmanabhan Seshaiyer
Since a permutation matrix is a matrix whose columns are a rearrangement of the columns of the identity matrix, the effect of multiplying by a permutation matrix P on the left is to rearrange the rows of A in the order in which the ejT appear in the rows of P. For example, 010001100123456789=456789123.
Stochastic Comparisons of Series and Parallel Systems with Kumaraswamy-G Distributed Components
Published in American Journal of Mathematical and Management Sciences, 2019
Consider a square matrix Π. It is said to be a permutation matrix if each row and column has exactly one entry unity and zeros elsewhere. Note, that there are such matrices of size n × n. For a permutation matrix Π that just interchanges two coordinates, a T-transform matrix is of the form , where and In is a identity matrix of order n. Let and be two T-transform matrices, where and are two permutation matrices that interchange two coordinates. Then, and have same structures if , otherwise they have different structures. An n × n matrix is said to be doubly stochastic if for each , and , and Next, we present some key concepts of multivariate majorization.