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Topic: Toeplitz matrix

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	Toeplitz Matrices, Algorithms and Applications
		Toeplitz matrices arise in many different theoretical and applicative fields, in the mathematical modelling of all the problems where some sort of shift invariance occurs in terms of space or of time.
		This shift invariance is reflected in the structure of the matrix itself where a south-eastern shift of the entries leaves the matrix unchanged.
		Toeplitz matrices may be finite or even infinite according to the features of the problem that is modelled.
	www.ercim.org /publication/Ercim_News/enw22/toeplitz.html (666 words)

	Toeplitz matrix - Wikipedia, the free encyclopedia
		In the mathematical discipline of linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant.
		These matrices have uses in computer science because it can be shown that the addition of two Toeplitz matrices can be done in O(n) time and the matrix multiplication of two Toeplitz matrices can be done in O(n log n) time.
		Toeplitz systems of form Ax = b can be solved by Levinson recursion.
	en.wikipedia.org /wiki/Toeplitz_matrix (250 words)

	Art Fresh : Article 'Matrix' (Site not responding. Last check: 2007-10-09)
		Orthogonal matrix - a matrix whose inverse is equal to its transpose, A−1 = AT.
		Similarity matrix Skew-Hermitian matrix - a square matrix which is equal to the negative of its conjugate transpose, A* = −A. Skew-symmetric matrix - a matrix which is equal to the negative of its transpose, AT = −A. Sparse matrix - containing mostly zeros Square matrix - an n by n matrix.
		Toeplitz matrix - a matrix with constant diagonals.
	www.art-fresh.net /DisplayArticle45184.html (1411 words)

	Matrix Reference Manual: Special Matrices
		A symmetric or Hermitian Hessenberg matrix is tridiagonal.
		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 (4421 words)

	PlanetMath: Toeplitz matrix
		This is version 5 of Toeplitz matrix, born on 2002-09-28, modified 2005-08-14.
		I am working with block Toeplitz matrices arising in the context of multivariate time-series data.
		In the case of block Toeplitz structures, suppose each block can be decomposed into at least one eigenvalue and associated eigenvector, then are the total number of eigenvalues in a block structure equivalent to the number of blocks necessarily?
	planetmath.org /encyclopedia/ToeplitzMatrix.html (271 words)

	Learn more about List of matrices in the online encyclopedia. (Site not responding. Last check: 2007-10-09)
		Diagonal matrix - All entries not on the main diagonal (the diagonal from the upper left to the lower right corner) are zero.
		Unitary matrix - A matrix whose conjugate transpose is its inverse.
		Permutation matrix - Matrix representation of a permutation.
	www.onlineencyclopedia.org /l/li/list_of_matrices.html (294 words)

	Hankel matrix - Wikipedia, the free encyclopedia
		The Hankel matrix is closely related to the Toeplitz matrix (a Hankel matrix is an upside-down Toeplitz matrix).
		The Hankel transform is the name sometimes given to the transformation of a sequence, where the transformed sequence corresponds to the determinant of the Hankel matrix.
		The Hankel transform of a sequence commutes with the binomial transform of a sequence.
	en.wikipedia.org /wiki/Hankel_matrix (161 words)

	TOEPLITZ Function
		The TOEPLITZ function generates a Toeplitz matrix from a vector, or a block Toeplitz matrix from a matrix.
		A block Toeplitz matrix has the property that all matrices on the diagonals are the same.
		The TOEPLITZ function is especially useful in time-series applications, where the covariance matrix of a set of variables with its lagged set of variables is often assumed to be a block Toeplitz matrix.
	www.jmu.edu /docs/sasdoc/sashtml/iml/chap17/sect243.htm (219 words)

	TOEPLITZ Function
		The TOEPLITZ function generates a Toeplitz matrix from a vector, or a block Toeplitz matrix from a matrix.
		A block Toeplitz matrix has the property that all matrices on the diagonals are the same.
		The TOEPLITZ function is especially useful in time-series applications, where the covariance matrix of a set of variables with its lagged set of variables is often assumed to be a block Toeplitz matrix.
	www.sfu.ca /sasdoc/sashtml/iml/chap17/sect243.htm (219 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Toeplitz matrices are square matrices with the property that each entry (except those in the first row and column) is equal to the entry immediately the northwest (previous col and previous row).
		Solution: sum yes, product no. That sum is Toeplitz easily follows because if C = A + B, then c_{i,j} = a_{i,j} + b_{i,j} // by def of matrix sum = a_{i-1,j-1} + b+{i-1,j-1} // by Toeplitz property of A,B = c{i-1,j-1} // by def of matrix sum.
		Since the first row,col of C is the componentwise sum of the first row,cols of A,B, and since C is Toeplitz as shown above, we see this gives the correct representation of the matrix sum.
	www.eecis.udel.edu /~saunders/courses/621/99f/p17b/p17.txt (399 words)

	gallery (MATLAB Function Reference)
		Explicit formulas are known for the inverse and determinant of a Cauchy matrix.
		A circulant matrix has the property that each row is obtained from the previous one by cyclically permuting the entries one step forward.
		The matrix is similar to the symmetric tridiagonal matrix with the same diagonal entries and with off-diagonal entries 1, via a similarity transformation with
	www.rrz.uni-hamburg.de /RRZ/Software/Matlab/HTML-Dokumentation.5-2/help/techdoc/ref/gallery.html (1295 words)

	Math Forum Discussions - Re: Eigenvalues of a random Hermitian Toeplitz matrix
		Lemma 4.1: Let tau_n, k be the eigenvalues of a Toeplitz matrix T_n(f).
		matrix is uniformly bounded over n if f is bounded.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?forumID=253&threadID=566871&messageID=1691530 (394 words)

	Toeplitz (DSP Blockset)
		The Toeplitz block generates a Toeplitz matrix from vectors defining the first column and first row.
		For example, the Toeplitz matrix generated from the input vector
		If you expect to generate code for the Toeplitz block using the Real-Time Workshop, you should ensure that inputs are contiguous in memory.
	www.tau.ac.il /cc/pages/docs/matlab/help/toolbox/dspblks/toeplitz.html (142 words)

	Advances in Computation: Theory and Practice, Book Series, Volume 4, Abstracts (Site not responding. Last check: 2007-10-09)
		After motivating the interest and the role of Toeplitz matrices by means of several examples of applications, we focus our interest on the solution of block banded block Toeplitz systems endowed with a two-level structure.
		The matrix structures dealt with are defined by means of singular displacement operators, and include, for example, Toeplitz, Toeplitz-plus-Hankel, and symmetric Loewner matrices.
		The flow starting from a matrix with a given displacement structure is enforced to preserve that structure, by adding a suitable constraint to the classical formulation.
	www.cs.okstate.edu /~actp/volumes/vol04abs.html (1865 words)

	Meet the Toeplitz matrix
		This covariance matrix is an example of a Toeplitz matrix.
		Toeplitz matrices are associated with elegant mathematics and rapid numerical solutions.
		Compared to Toeplitz methods of solving equation (19), the CG method of solving (18) is slower.
	sepwww.stanford.edu /ftp/prof/pvi/tsa/paper_html/node20.html (379 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - Matrices
		The ESSL subroutines adjust their selection of elements from the matrix when an argument in the calling sequence indicates that the transposed matrix is to be used in the computation.
		When a symmetric matrix is stored in lower storage mode, the lower triangular part of the symmetric matrix is stored, including the diagonal, in a two-dimensional array.
		A sparse matrix is a matrix having a relatively small number of nonzero elements.
	www.ncsa.uiuc.edu /UserInfo/Resources/Hardware/IBMp690/IBM/usr/lpp/essl.html.en_US/html/essl43.html (8125 words)

	MAT520 - Project 1
		grcar Grcar matrix -- a Toeplitz matrix with sensitive eigenvalues.
		hanowa Matrix whose eigenvalues lie on a vertical line in the complex plane.
		ris Ris matrix -- a symmetric Hankel matrix.
	math.la.asu.edu /~bdw/MAT520/Fa98/P1 (175 words)

	: Class RectBandedMatrix
		The representation of the matrix in the container is the following: k entries of the first row (from the left to the right), k entries of the second row, and so on.
		A special case of a matrix, a Toeplitz matrix, is also implemented.
		Here a and b are scalar multipliers, A is a symmetric banded matrix, H is a rectangular banded matrix (contents of this object), P is a weight diagonal matrix.
	www.sscc.ru /matso/rozhenko/applib/api/ru/sscc/matrix/RectBandedMatrix.html (480 words)

	Citations: Gaussian elimination with partial pivoting for structured matrices - Gohberg, Kailath, Olshevsky ...
		They converted indefinite Toeplitz and Hankel matrices to Cauchy like matrices using fast trigonometric transforms such as the discrete Fourier transform (DFT) The displacement structure of Cauchy like matrices is invariant to permutation.
		The problem with this method is that any real valued Toeplitz matrix is converted to a complex Cauchy type matrix and the entire factorization algorithm proceeds in complex arithmetic.
		Symmetric positive definite block Toeplitz matrices In this section we present a block generalization of the classical Schur algorithm [8, 9] using block hyperbolic Householder reflectors.
	citeseer.ist.psu.edu /context/324933/0 (1390 words)

	Block Toeplitz Matrices
		These are not necessarily scalar Toeplitz matrices, but, similarly to those, the block structure can be used in a fast matrix-vector multiplication.
		Since we know how this is done for one circulant matrix it is now easily generalized to a matrix consisting of many circulant blocks.
		The dimension of the blocks (n) was as close as possible to the block dimension (N) of the matrix, while still keeping both n and N a power of two.
	www.mai.liu.se /~evlun/pub/lic/node11.html (1230 words)

	Guide and Reference
		> 0.0, the input Toeplitz matrix is indefinite (that is, it is not positive definite and it is not negative definite).
		For CWLEV and ZWLEV, matrix A is a complex Hermitian Toeplitz matrix whose first row is represented by vector x.
		It uses the same input Toeplitz matrix as in Example 2, which is not positive definite.
	www.cs.unb.ca /docs/spdocs/essl/essl420.html (1026 words)

	[No title]
		A Toeplitz matrix is defined by EMBED Equation.3 , which means that all entries on the same downward diagonal are required to be the same.
		Here is an example of a Hankel matrix: EMBED Equation.3 A Vandermonde matrix begins with a column vector, and raises the entries of the column to the c-1 power, where c is the number of the column in the matrix.
		MNEG: Input: Matrix M Output: Additive inverse of M The sum of M and the additive inverse of M is 0.
	www.cs.caltech.edu /~joyjoy/CSReport.doc (2247 words)

	Toeplitz (DSP Blockset)
		The Toeplitz block generates a Toeplitz matrix from inputs defining the first column and first row.
		When the Symmetric check box is selected, the block generates a symmetric (Hermitian) Toeplitz matrix from a single input,
		Also see Matrix Operations for a list of all the blocks in the Matrix Operations library.
	www.weizmann.ac.il /matlab/toolbox/dspblks/toeplitz.html (203 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Toeplitz matrices of dimenion n × n can be inverted with O(n
		We construct a 4 × 4 Toeplitz matrix with 3 bands:
		We construct a 3 × 5 Toeplitz matrix with symbolic entries:
	www.sciface.com /STATIC/DOC30/eng/linalg_toeplitz.html (99 words)

	MuPAD documentation
		Just make sure that both the Toeplitz entries t as well as the components of the ’right hand side’ y are elements of the desired coefficient ring.
		The Toeplitz entries t and the right hand side y of the linear system are entered as row vectors:
		We generate the corresponding Toeplitz matrix and use a generic linear solver such as
	www.mupad.de /doc/30/eng/linalg_toeplitzSolve.html (401 words)

	Toeplitz (DSP Blockset)
		The Toeplitz block generates a Toeplitz matrix from inputs defining the first column and first row.
		Col = [1 2 3 4 5] Row = [7 7 3 3 2 1 3]
		When the Symmetric check box is selected, the block generates a symmetric (Hermitian) Toeplitz matrix from a single input,
	www.mathworks.com /access/helpdesk_r13/help/toolbox/dspblks/toeplitz.html (238 words)

	: Class SymBandedMatrix
		A banded matrix is stored in the container by its upper half values.
		The ordering of the matrix values in the container is the following: (k+1) upper diagonal entries of the first row beginning from the diagonal entry, (k+1) upper diagonal entries of the second row, and so on.
		A number of nonempty diagonals in upper half of matrix (the main diagonal is also counted).
	www.sscc.ru /matso/rozhenko/applib/api/ru/sscc/matrix/SymBandedMatrix.html (472 words)

	Matrix Reference Manual: Matrix Identities
		These identities are useful because it says how a matrix changes if you add a bit onto its inverse.
		toeplitz matrix, it is possible to exchange the components of the vector and matrix.
		The matrix on the right is the sum of a lower triangular toeplitz and an upper triangular hankel matrix and has 2b as its first column.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/identity.html (328 words)

	LINPLUS - Linear Algebra Utilities
		The routines are primarily organized by the storage format of the matrix they operate on.
		CTO_SL solves the complex Toeplitz system A * X = B. copies a complex Toeplitz matrix into a general matrix.
		SPB_FA factors a positive definite symmetric band matrix A. multiplies a vector times a matrix that was factored by SPB_FA.
	orion.math.iastate.edu /burkardt/f_src/linplus/linplus.html (2124 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Most of the functions take an input argument % that specifies the order of the matrix, and unless otherwise % stated, return a single output.
		% grcar Grcar matrix -- a Toeplitz matrix with sensitive eigenvalues.
		% ris Ris matrix -- a symmetric Hankel matrix.
	www.clemson.edu /cle4_share/CWE/COES0915_CLUG/REFERENCE/matlabr14/toolbox/matlab/elmat/gallery.m (471 words)

	[No title]
		Iteration limit could fail under certain condition Bug 09.11.03 Switch off/on the determinat computing in the GJ() Gauss-Jordan routine This avoid the determinant overflow error during the matrix inverse computing Now also this addin has a little menu bar "Matrix Tool" performing useful matrix operations: selecting, coping piece of matrix, and running macros.
		The following new macros are added: Matrix Selector - Selects special matrix format: triangular, diagonal, tridiagonal Matrix Scarp Paster - Pastes matrix pieces in the worksheet Matrix Generator - Generates several kind of matrices: random, tartaglia, hilbert, etc. ---------------------------------------------------------------------------------- MATRIX.XLA- Ver.
		Now return also the permuation matrix P, being A = PLU MatNormalize bug fixed: uncorrected value returned M_DET accuracy improved for integer matrices M_INV accuracy improved for integer matrices SYSLIN accuracy improved for integer matrices M_EXP efficence improved with the Padè approximation (thanks to Gregory Klein) ---------------------------------------------------------------------------------- MATRIX.XLA - Ver.
	www.msu.edu /course/fw/877/bence/matrix_1.8/matrix_review.txt (399 words)

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