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Topic: Permutation matrix

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	Permutation matrix - Wikipedia, the free encyclopedia
		The trace of a permutation matrix is the number of fixed points of the permutation.
		corresponding to the permutation π=(1)(2 4 5 3) is
		is the permutation form of the permutation matrix.
	en.wikipedia.org /wiki/Permutation_matrix (764 words)

	Matrix Reference Manual: Special Matrices
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		A Rotation matrix is orthogonal with a determinant of +1.
		A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4470 words)

	PlanetMath: example of permutation matrix
		(first column from identity matrix is the second column, second column of identity matrix is fourth column, etc).
		"example of permutation matrix" is owned by drini.
		This is version 1 of example of permutation matrix, born on 2005-02-17.
	planetmath.org /encyclopedia/ExampleOfPermutationMatrix.html (78 words)

	The Best Page In The Universe? - Ken "Reaper of Souls" Orji (Site not responding. Last check: 2007-10-11)
		The determinant of a matrix A is denoted det(A) or A.
		If this permutation is a sequence of an even number of transpositions, it is impossible to write this permutation as a sequence of an odd number of transpositions.
		Let B be the matrix formed by replacing in A the ith row with the jth row, leaving the jth row unchanged.
	coweb.math.gatech.edu:8888 /calculus/3205 (1924 words)

	Matrix Reference Manual
		Thus [1 2 3; 4 5 6] is a matrix with 2 rows and 3 columns.
		ADJ(A) is the adjoint of the square matrix A.
		DIAG(a) is the diagonal matrix whose diagonal elements are the elements of a.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/intro.html (1339 words)

	Matrix Manual: Special Matrices
		matrix, A, is an nn Toeplitz matrix* in which a(i,j) is a function of {(i-j) modulo n}.
		A is symmetric iff it is congruent to a diagonal matrix.
		Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
	www.psi.toronto.edu /matrix/special.html (2192 words)

	[No title]
		The matrix M must have entries in the ring of integers, or the ring of Gaussian integers.
		p[i, j] is the identity matrix with the i-th and j-th columns interchanged.
		The field is determined by the coefficients of the polynomials in the input matrix.
	www.willamette.edu /~zizza/Software/ElementaryDecompositions.m (513 words)

	Dr. Dobb's \| Permutation Generation Using Matrices \| July 22, 2001
		A "cyclic-permutation matrix" is an nxn matrix whose first row is a pivot permutation and whose ith row is obtained by a full-right rotate of the (i-1)th row for i=2,...,n.
		The cyclic-permutation matrix can be created using full-left rotates, but the matrix will have to be created bottom up; that is, the pivot permutation is the nth row of the matrix, and for i=(n-1),...,1, the ith row is created by performing a full-left rotate on the (i+1)th row.
		In the cyclic-permutation matrix, the primary diagonals or the secondary diagonals, depending on the direction of rotates used (primary for right and secondary for left), have the same number.
	www.ddj.com /184409671 (1716 words)

	Encryption Algorithms and Permutation Matrices
		Permutations 're-arrange' the elements of the alphabet A. The fact that a permutation is a bijection ensures that it has an inverse that returns each character to its original location.
		permutation matrix is a square matrix of zeros and exactly N ones such that there is exactly 1 one per column or row.
		As an example of a non-symmetric permutation matrix, let C be the permutation given by the matrix
	www.haikulabs.com /encrypt.htm (1410 words)

	Paulus Gerdes (Site not responding. Last check: 2007-10-11)
		The inverse permutation of (1 4 3 2 5) is (5 2 3 4 1) that may be written as (1 5 2 3 4), and its representation is the vertical mirror image of the first (Figure 10).
		Symmetrical representations of the permutations (1 2 4 5 3) and (1 3 5 4 2)
		Figures 13 and 14 illustrate the polygonal-circle representations of permutation matrices that are positive alternating cycle matrices in the cases m=7 and m=10.
	www.mi.sanu.ac.yu /vismath/gerd2006first/index.html (2259 words)

	NCLDUDecomposition[aMatrix, Options]
		, and the fourth is the permutation matrix
		If all the diagonal elements are zero, then a permutation matrix is used to place a nonzero entry in the (2,1) position which will automaticaly place a nonzero entry in the (1,2) position if the matrix is symmetric.
		pivoting strategy assumes that the matrix is symmetric in that it only ensures that the (2,1) entry is nonzero (assuming by symmetry that the (1,2) is also zero).
	math.ucsd.edu /~ncalg/NCBIGDOC02/node64.html (565 words)

	Permutation Equivalent Codes
		Remark: It is often convenient to express the permutation between codes in a permutation matrix, which is a matrix with exactly one
		Now choose a permutation of the columns of the new generator matrix so that these columns are moved to the order that produces
		The ``stabilizer" of the code is the set of all elements of the symmetric group which fix the code (as a set), i.e., the permutation automorphism group of the code.
	www.math.unl.edu /~s-bbockel1/coding-notes/node7.html (618 words)

	Elementary Row Operations
		This permutation is applied to the rows or columns in em and me respectively.
		One permutation describes the term from det(c), and the other is the magic numbers we use to find a subterm within the expansion of this term.
		This permutation is joined to the term of det(b), and the result is the permutation that arranges rows and columns in det(c).
	www.mathreference.com /la-det,erop.html (1198 words)

	AMTH247 Lecture 9 Linear Equations III
		This is the same upper triangular matrix which was computed by Scilab at the end of Lecture 8.
		The effect of partial pivoting is to multiply by a permutation matrix before multiplying by the elimination matrix at each step.
		is not lower triangular, however it is a row permutation of a lower triangular matrix.
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_09/lecture (530 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-11)
		A border banded matrix is a 2 by 2 block matrix comprising a (large) leading block which is a square banded matrix, two dense rectangular side strips, and a (small) trailing block which is a square dense matrix.
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13553 words)

	Combinatorial Geometry of Birkhoff Polytopes
		Recall, that multiplying a matrix by a permutation matrix shuffles columns and rows.
		Since the matrix is doubly stochastic, the right side of the equation is 1 subtracted by the other elements in the row.
		Recall, the equations with ones represent the permutation, by pivoting one of these equations, the permutation being considered is changed.
	linus.highpoint.edu /~zjones/final (3269 words)

	Euler angles from matrix - CGAFaq
		For an even permutation of the axes, the columns of P are also an even permutation of the identity matrix columns; hence P has determinant +1.
		For an odd permutation of the axes, such as (x,y,z) ↦ (z,y,x), achieved by a single swap x ↔ z, the columns of P are an odd permutation of the identity matrix columns,
		Common code can extract angles for any permutation by permuting indices; however, because odd permutations reverse the sense of rotation, the extracted angles must then be negated.
	cgafaq.info /wiki/Euler_angles_from_matrix (1013 words)

	GNU Scientific Library -- Reference Manual: LU Decomposition
		where P is a permutation matrix, L is unit lower triangular matrix and U is upper triangular matrix.
		The permutation matrix P is encoded in the permutation
		The j-th column of the matrix P is given by the k-th column of the identity matrix, where k = p_j the j-th element of the permutation vector.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_203.html (400 words)

	ZPR: Print a Matrix or Permutation
		This program reads a matrix or permutation(s) from Datafile (G1 by default), and outputs the object(s) therein to Textfile in a format suitable for reading by a human, or by the ZCV program.
		For a permutation they are `1', number of points, and number of permutations.
		For example, to convert a permutation to a matrix, use ZPR and then edit the type from 12 to 2.
	www.math.rwth-aachen.de /~MTX/htmldoc/node20.html (401 words)

	zjadrp
		M(input) On entry, M specifies the number of rows in the matrix A. Unchanged on exit.
		K(input) On entry, K specifies the number of columns in the matrix A. Unchanged on exit.
		On exit, INDX is is overwritten by the column indices of the output permuted jagged diagonal matrix.
	docs.sun.com /source/819-0497/zjadrp.html (432 words)

	problemset-june08
		is an invertible matrix and only permutes the columns of the generator matrix of
		is a permutation matrix, since all but the last column have precisely one 1 in them, and similarly all but the last row have precisely one 1 in them, since
		A generating matrix for a punctured code is obtained by deleting the corresponding columns from the original code's generating matrix.
	www.math.unl.edu /~s-bbockel1/problemset-june08 (329 words)

	Plane Partitions and Their Connection to the Alternating Sign Matrix Conjecture
		An alternating sign matrix(ASM) is a matrix of 0's, 1's, and -1's with constant row sums and column sums, both equal to 1.
		matrix whose only entry is the determinant of the initial matrix.
		is associated with the permutation matrix with a one in place of terms
	www-math.cudenver.edu /~rrosterm/combproj/combproj.html (1504 words)

	GNU Scientific Library -- Reference Manual - LU Decomposition (Site not responding. Last check: 2007-10-11)
		where @math{P} is a permutation matrix, @math{L} is unit lower triangular matrix and @math{U} is upper triangular matrix.
		The permutation matrix @math{P} is encoded in the permutation
		The @math{j}-th column of the matrix @math{P} is given by the @math{k}-th column of the identity matrix, where @math{k = p_j} the @math{j}-th element of the permutation vector.
	www.math.utah.edu:8080 /software/gsl/gsl-ref_203.html (419 words)

	PA = LU Factorization with Pivoting
		We have seen in Example 3 an example of a nonsingular matrix A could not be directly factored as A = LU.
		If row interchanges are permitted then a factorization of a "permuted matrix" will be obtained.
		An n×n permutation matrix P is a matrix with precisely one entry whose value is "1" in each column and row, and all of whose other entries are "0." The rows of P are a permutation of the rows of the identity matrix and P can be written as
	math.fullerton.edu /mathews/n2003/LUFactorMod.html (469 words)

	[No title]
		The second value of options->Fact assumes that a matrix with the same * sparsity pattern as A has already been factored: * * - options->Fact = SamePattern: A is factored, assuming that it has * the same nonzero pattern as a previously factored matrix.
		This is because the * previous column permutation from ScalePermstruct->perm_c is used as * input.
		This is because the permutations from ScalePermstruct->perm_r * and ScalePermstruct->perm_c are used as input.
	acts.nersc.gov /tau/programs/pdgssvx/pdgssvx.c (2861 words)

	LU Decomposition - GNU Scientific Library -- Reference Manual
		A general square matrix A has an LU decomposition into upper and lower triangular matrices,
		The lower triangular part of the input matrix (excluding the diagonal) contains L. The diagonal elements of L are unity, and are not stored.
		It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtain the same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for details).
	www.gnu.org /software/gsl/manual/html_node/LU-Decomposition.html (592 words)

	GNU Scientific Library -- Reference Manual - Applying Permutations (Site not responding. Last check: 2007-10-11)
		, considered as a row-vector acted on by a permutation matrix from the right, @math{v' = v P}.
		The @math{j}-th column of the permutation matrix @math{P} is given by the @math{p_j}-th column of the identity matrix.
		Note that for permutation matrices the inverse is the same as the transpose.
	www.math.utah.edu:8080 /software/gsl/gsl-ref_184.html (133 words)

	lu
		as the product of two essentially triangular matrices, one of them a permutation of a lower triangular matrix and the other an upper triangular matrix.
		Most of the algorithms for computing LU factorization are variants of Gaussian elimination.
		An upper triangular matrix that is a factor of
	grove.ufl.edu /matlab_help/techdoc/ref/lu.html (209 words)

	Permutation and Matrix Groups
		For an overview of computational methods for permutation groups see the book [Se03] of Ákos Seress.
		there are also special methods in the GAP library and a private GAP4 package matrixss implements a Schreier-Sims algorithm for matrix groups, including both the standard deterministic and the standard probabilistic approach.
		Routines to recognize isomorphism types of matrix groups using the Aschbacher classification (which recently have been in the focus of attention in computational group theory) are still (2005) only available through the GAP 3 package matrix.
	www-groups.dcs.st-and.ac.uk /~gap/Overview/Capabilities/permgroups.html (255 words)

	LinBox: TemplateLinBox::Permutation< Storage > class Reference (Site not responding. Last check: 2007-10-11)
		Constructor from a vector of indices This constructor creates a permutation matrix based on a vector of indices
		Constructor from a dimension This constructor creates an n x n permutation matrix, initialized to be the identity
		The documentation for this class was generated from the following file:
	www.eecis.udel.edu /~linbox/doc/html/classLinBox_1_1Permutation.html (65 words)

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