PropPermMatrices - Let B = P(r,s A Then B is the same as A...

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Page 1 of 1 5/14/2003 PropPermMatrices.doc Properties of Permutation Matrices (last rev 4/28/03) Let I denote the identity matrix of order n. Let P (r,s) be the I matrix of order n with rows r and s interchanged. The P (r,s) above is called an elementary permutation matrix. Let P k = P k (r k ,s k ) denote one of a series of elementary permutation matrices for k=1,...,K. A general permutation matrix, P , is a product of a series of elementary permutation matrices: P = P 1 P 2 ... P K Let A be square of order n. Let R 1 , R 2 , ..., R n denote the rows of A . Let C 1 , C 2 , ... , C n denote the columns of A. Property 1 : Let A be a square matrix of order n and P (r,s) be an elementary permutation matrix of order n.
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Unformatted text preview: Let B = P (r,s) A . Then B is the same as A except that R r and R s are interchanged. I.e. Premultiplication swaps rows. Property 2 : Let A be a square matrix of order n and P (r,s) be an elementary permutation matrix of order n. Let B = AP (r,s). Then B is the same as A except that C r and C s are interchanged. I.e. Postmultiplication swaps columns. Property 3 : The inverse of an elementary permutation matrix is the elementary permutation matrix; i.e. P-1 (r,s)= P (r,s) Property 4 : The inverse of a general permutation matrix is its transpose; i.e. P-1 = P T ....
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