59 lines
2.1 KiB
Matlab
59 lines
2.1 KiB
Matlab
% PERMUTATION : Helpful notes on permutation matrices and vectors.
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%
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% There are two ways to represent permutations in Matlab.
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%
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% A *permutation matrix* is a square matrix P that is zero except for exactly
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% one 1 in each row and each column. A *permutation vector* is a 1 by n
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% or n by 1 vector p that contains each of the integers 1:n exactly once.
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%
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%
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% Let A be any n by n matrix, P be an n by n permutation matrix, and p be
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% a permutation vector of length n. Then:
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%
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% P*A is a permutation of the rows of A.
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% A*P' is the same permutation of the columns of A. Remember the transpose!
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%
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% A(p,:) is a permutation of the rows of A.
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% A(:,p) is the same permutation of the columns of A.
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%
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% Matrix P and column vector p represent the same permutation if and only if
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% any of the following three equivalent conditions holds:
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%
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% p = P*(1:n)' or P = sparse(1:n,p,1,n,n) or P = I(p,:),
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%
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% where I = speye(n,n) is the identity matrix of the right size.
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%
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% The inverse of a permutation matrix is its transpose:
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%
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% Pinv = inv(P) = P'
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%
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% The inverse of a permutation vector can be computed by a one-liner:
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%
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% pinv(p) = 1:n;
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%
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% (This works only if pinv doesn't already exist, or exists with the correct
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% dimensions. To be safe, say: pinv=zeros(1,n); pinv(p)=1:n; )
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%
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% Products of permutations go one way for matrices and the other way for
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% vectors. If P1, P2, and P3 are permutation matrices and p1, p2, and p3
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% are the corresponding permutation vectors, then
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%
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% P1 = P2 * P3 if and only if p1 = p3(p2).
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%
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% Permutation vectors are a little more compact to store and efficient
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% to apply than sparse permutation matrices (and much better than full
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% permutation matrices). A sparse permutation matrix takes about twice
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% the memory of a permutation vector. The Matlab function LU returns
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% a permutation matrix (sparse if the input was sparse); most other
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% built-in Matlab functions return permutation vectors, including
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% SYMRCM, SYMMMD, COLAMD, DMPERM, and ETREE.
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% John Gilbert, 6 April 1995
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% Copyright (c) 1995 by Xerox Corporation. All rights reserved.
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help permutation
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