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Topic: Triangular matrix

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	PlanetMath: triangular matrix
		In fact, triangular matrices are so useful that much computational linear algebra begins with factoring (or decomposing) a general matrix or matrices into triangular form.
		The eigenvalues of a triangular matrix are the diagonal elements.
		This is version 5 of triangular matrix, born on 2002-01-16, modified 2004-03-12.
	planetmath.org /encyclopedia/LowerTriangularMatrix.html (249 words)

	Triangular matrix - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-06)
		A triangular matrix with zero entries on the main diagonal is strictly upper or lower triangular.
		The transpose of a upper triangular matrix is a lower triangular matrix and vice versa.
		The set of 2 by 2 triangular matrices is called the parabolic subgroup; 3 by 3 and larger normed triangular matrices form the Heisenberg group.
	en.wikipedia.org /wiki/Triangular_matrix (496 words)

	PlanetMath: strictly upper triangular matrix
		A strictly upper triangular matrix is an upper triangular matrix which has 0 on the main diagonal.
		Similarly A strictly lower triangular matrix is an upper triangular matrix which has 0 on the main diagonal.
		This is version 4 of strictly upper triangular matrix, born on 2003-06-19, modified 2003-07-10.
	www.planetmath.org /encyclopedia/StrictUpperTriangularMatrix.html (93 words)

	LU decomposition - Wikipedia, the free encyclopedia
		In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix.
		is again a lower triangular matrix, and the multiplication of two lower triangular matrices is again a lower triangular matrix, it follows that L is a lower triangular matrix.
		Note that in both cases we have triangular matrices (lower and upper) which can be solved directly using forward and backward substitution without using the Gaussian elimination process (however we need this process or equivalent to compute the LU decomposition itself).
	en.wikipedia.org /wiki/LU_decomposition (704 words)

	Guide and Reference
		For a description of triangular matrices and how they are stored in upper- and lower-triangular storage mode and in upper- and lower-triangular-packed storage mode, see "Triangular Matrix".
		Example 1: This example shows how the inverse of matrix A is computed, where A is a 5 by 5 upper triangular matrix that is not unit triangular and is stored in upper-triangular storage mode.
		Example 2: This example shows how the inverse of matrix A is computed, where A is a 5 by 5 lower triangular matrix that is unit triangular and is stored in lower-triangular storage mode.
	www.umiacs.umd.edu /~jhu/DOCS/SP/docs/essl/essl341.html (753 words)

	Triangular matrix (Site not responding. Last check: 2007-11-06)
		In the mathematical discipline of linear algebra, a triangular matrix is a special kind of matrix where the entries below or above the main diagonal are zero.
		The identity matrix is a normed upper and lower triangular matrix.
		Because triangular matrices are easy to calculate they are very important in numerical analysis.The LU decomposition gives an algorithm to decompose any invertible matrix A into a normed upper triangle matrix L and a normed lower triangle matrix R.
	www.sciencedaily.com /encyclopedia/triangular_matrix (318 words)

	Serializer/deserializer with a triangular matrix - Patent 5101202
		The matrix cells are arranged according to a triangle, the cells being arranged one with respect to the other according to the structural corresponding to folding a square matrix along its diagonal.
		an n-rows and n-columns matrix of 1-bit registers, each of which is connected through its input to a first switch connected to the output of the register in the same row and lower rank column and to a second switch connected to the output of the register in the same column and upper rank row.
		The triangular structure according to the invention may seem surprising because, when blocks are placed in an integrated circuit, each one is more or less inscribed within a rectangle and the unused portion of a triangular block is liable to be lost.
	www.freepatentsonline.com /5101202.html (2530 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - STRMV, DTRMV, CTRMV, ...
		Matrix A is a complex 4 by 4 upper triangular matrix that is unit triangular, stored in upper-triangular storage mode.
		Matrix A is a real 4 by 4 upper triangular matrix that is not unit triangular, stored in upper-triangular-packed storage mode.
		Matrix A is a complex 4 by 4 upper triangular matrix that is unit triangular, stored in upper-triangular-packed storage mode.
	www.nersc.gov /vendor_docs/ibm/essl/essl110.html (1062 words)

	Guide and Reference (Site not responding. Last check: 2007-11-06)
		A unit triangular matrix is a triangular matrix in which all the diagonal elements have a value of one; that is:
		When an upper-triangular matrix is stored in upper-triangular-packed storage mode, the upper triangle of the matrix is stored, including the diagonal, in a one-dimensional array.
		When a lower-triangular matrix is stored in lower-triangular-packed storage mode, the lower triangle of the matrix is stored, including the diagonal, in a one-dimensional array.
	www.pdc.kth.se /doc/SP/manuals/essl/html/essl138.html (580 words)

	Guide and Reference (Site not responding. Last check: 2007-11-06)
		Matrix A is a real 4 by 4 lower unit triangular matrix, stored in lower-triangular storage mode.
		Matrix A is a complex 4 by 4 upper unit triangular matrix, stored in upper-triangular storage mode.
		Matrix A is a complex 4 by 4 upper unit triangular matrix, stored in upper-triangular-packed storage mode.
	www.pdc.kth.se /doc/SP/manuals/essl/html/essl339.html (1108 words)

	psi^3 as an upper triangular matrix, by Jonathan Barker and Victor Snaith (Site not responding. Last check: 2007-11-06)
		psi^3 as an upper triangular matrix, by Jonathan Barker and Victor Snaith
		In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu \wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic integers.
		We identify the conjugacy class of the matrix corresponding to 1 \wedge \psi^3, where \psi^3 is the Adams operation.
	www.math.uiuc.edu /K-theory/0728 (82 words)

	Special Matrices: Triangular, Symmetric, Diagonal (Site not responding. Last check: 2007-11-06)
		We have seen that a matrix is a block of entries or two dimensional data.
		A symmetric matrix is a matrix equal to its transpose.
		A diagonal matrix is a symmetric matrix with all of its entries equal to zero except may be the ones on the diagonal.
	www.sosmath.com /matrix/matrix3/matrix3.html (345 words)

	Triangular matrix (Site not responding. Last check: 2007-11-06)
		Algebras of upper triangular matrices have a natural generalisation in functional analysis which yields nest algebras.
		The determinant of a triangular matrix equals the product of the diagonal entries.
		Category:Numerical linear algebra Category:Matrices de:Dreiecksmatrix es:Matriz triangular fr:Matrice triangulaire it:Matrice triangolare ja:&19977;&35282;&34892;&21015; pl:Macierz tr&243;jk&261;tna
	triangular-matrix.area51.ipupdater.com (694 words)

	Extract Triangular Matrix (DSP Blockset)
		Extract the lower or upper triangle from an input matrix.
		The Extract Triangular Matrix block creates a triangular matrix output from the upper or lower triangular elements of an M-by-N input matrix.
		The first row of the output matrix is therefore identical to the first row of the input matrix.
	www.weizmann.ac.il /matlab/toolbox/dspblks/extracttriangularmatrix.html (251 words)

	Guide and Reference
		For unit triangular matrices, the elements of the diagonal are assumed to be one.
		When using an upper or lower triangular matrix, the unreferenced elements in the strictly lower or upper triangular part, respectively, are assumed to be zero.
		After the global matrix X is distributed over the process grid, only a portion of the global data structure is used--that is, global vector x, which is a column-distributed vector.
	www.umiacs.umd.edu /~jhu/DOCS/SP/docs/pessl/pessl133.html (1628 words)

	Dual-tone multi-frequency - Wikipedia, the free encyclopedia
		Today DTMF is used for most call setup to the telephone exchange, at least in the Western world, and trunk signalling is now done out of band using the SS7 signaling system.
		The trunk signalling tones were different from the tones known as touch tone with a triangular matrix being used rather than a square matrix.
		The DTMF keypad is laid out in a 4×4 matrix, with each row representing a low frequency, and each column representing a high frequency.
	en.wikipedia.org /wiki/Dual_tone_multi_frequency (942 words)

	Similarity over C
		Thus, every matrix is similar to a matrix in Jordan form, and two matrices in Jordan form are similar if and only if they have the same blocks, although not necessarily arranged in the same order.
		One can write a matrix in Jordan form as the sum of a diagonal matrix D with the same diagonal entries, and an upper triangular matrix N with all 0's on the diagonal.
		There is an algorithmic process for finding a matrix S such that S^{-1} A S is in Jordan form, but we shall not give the details here: the method, along with an explanation of uniqueness, can be found in any text for a second course in linear algebra.
	www.math.lsa.umich.edu /~hochster/419/cxsim.html (1175 words)

	HLAPack.DT (Site not responding. Last check: 2007-11-06)
		DTBCON estimates the reciprocal of the condition number of a triangular band matrix A, in either the 1-norm or the infinity-norm.
		A, B are upper quasi triangular and D, E are upper triangular.
		DTRCON estimates the reciprocal of the condition number of a triangular matrix A, in either the 1-norm or the infinity-norm.
	www.isi.edu /~hdaume/HBlas/docs/HLAPack.DT.html (2248 words)

	GaussianElimination
		Because the first step can be seen abstractly as factoring the matrix of coefficients into the product of a lower triangular and an upper triangular matrix, it is often referred to as LU factorization.
		This data object represents the factorization of the original matrix together with information regarding any permutation of the rows that might be necessary to maintain numerical stability in the computation.
		The zeros in the lower part of the upper triangular matrix, the upper part of the lower triangular matrix, and the ones along the diagonal of the lower triangular matrix are not stored explicitly.
	documents.wolfram.com /v4/AddOns/StandardPackages/LinearAlgebra/GaussianElimination.html (393 words)

	LU Decomposition
		The upper triangular matrix results from reducing the original matrix into row echelon form by way of elementary row operations(excluding row interchange).
		Notice that this is a square matrix and is not yet augmented with the vector of constants from the equations.
		In fact nothing is being done to the upper triangular matrix, but we need the 1 in the lower matrix to show that.
	www.aspire.cs.uah.edu /textbook/LU.html (650 words)

	Extract Triangular Matrix (DSP Blockset)
		The Extract Triangular Matrix block creates a triangular matrix output from the upper or lower triangular elements of a rectangular input matrix.
		The upper triangular elements of the output are zero.
		If you expect to generate code for the Extract Triangular Matrix block using the Real-Time Workshop, you should ensure that inputs are contiguous in memory.
	www.math.muni.cz /matlab/toolbox/dspblks/extracttriangularmatrix.html (174 words)

	Triangular matrix - InfoSearchPoint.com (Site not responding. Last check: 2007-11-06)
		In mathematics, a triangular matrix is a matrix with entries that have elements either completely above (upper triangular) or completely below (lower triangular) the principal diagonal.
		1 & 4 & 2 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \\ \end{pmatrix} is upper triangular and
		1 & 0 & 0 \\ 2 & 8 & 0 \\ 4 & 9 & 7 \\ \end{pmatrix} is lower triangular.
	www.infosearchpoint.com /display/Triangular_matrix (108 words)

	Matrix Market: Glossary
		A matrix is diagonally dominant if the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in its row (or column).
		A Hessenberg matrix is `almost' triangular, that is, it is (upper or lower) triangular with one additional off-diagonal band (immediately adjacent to the main diagonal).
		An ill-conditioned matrix is one where the solution to Ax=b is overly sensitive to perturbations in A or b.
	math.nist.gov /MatrixMarket/glossary.html (756 words)

	Matrix Math (Site not responding. Last check: 2007-11-06)
		When you are done, the original matrix >should be the identity matrix, and the original identity matrix will be the >inverse.
		A lower triangular matrix is a square matrix having only >zeroes above its main diagonal.
		A triangular matrix is either upper or >lower triangular.
	www.newton.dep.anl.gov /askasci/math99/math99083.htm (545 words)

	MATRIZ TRIANGULAR (Site not responding. Last check: 2007-11-06)
		En matemáticas, una matriz triangular es una matriz cuyos elementos no nulos se encuentran todos encima de la diagonal principal (triangular superior) o debajo de la misma (triangular inferior).
		1 and 0 and 0 \\ 2 and 8 and 0 \\ 4 and 9 and 7 \\ \end{pmatrix} $es$ triangular inferior.
		Se suelen emplear las letras U y L, respectivamente, ya que U es la inicial de "upper triangular matrix" y L de "lower triangular matrix", los nombres que reciben estas matrices en inglés.
	www.espnuevomilenio.org /encyclopedia/M/Matriz_triangular (139 words)

	Operations on a Triangular Matrix (Site not responding. Last check: 2007-11-06)
		This section deals with lower triangular matrices, but the same results hold for upper triangular matrices.
		Use the previous procedure to invert a lower triangular matrix.
		The lower triangular matrices form a subgroup of the nonsingular matrices, and the lower triangular matrices with determinant 1 form a smaller subgroup, and the lower triangular matrices with ones down the main diagonal form an even smaller subgroup.
	www.mathreference.com /la-det,triop.html (180 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		STRSM solves one of the matrix equations where alpha is a scalar, X and B are m by n matrices, A is a unit, or non-unit, upper or lower triangular matrix and op(A) is one of op(A) = A or op(A) = A'.
		The matrix X is overwritten on B. On entry, SIDE specifies whether op(A) appears on the left or right of X as follows: SIDE = 'L' or 'l' op(A)X = alphaB. SIDE = 'R' or 'r' Xop(A) = alphaB. Unchanged on exit.
		Before entry with UPLO = 'L' or 'l', the leading k by k lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced.
	www.ibiblio.org /gferg/ldp/man/manl/strsm.l.html (521 words)

	Triangular Matrix (Site not responding. Last check: 2007-11-06)
		Although triangular facets are always planar, quadrangles in space generally are not...
		In linear algebra, a triangular matrix is a special kind of matrix where the entries below or above the main diagonal are zero.
		A matrix equation with a normed upper triangular matrix R can be solved in an analogous way.
	triangular-matrix.wikiverse.org (371 words)

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