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Topic: Lower triangular matrix

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Upper triangular matrix

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	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 a lower triangular matrix which has 0 on the main diagonal.
		This is version 5 of strictly upper triangular matrix, born on 2003-06-19, modified 2006-06-22.
	planetmath.org /encyclopedia/StrictUpperTriangularMatrix.html (101 words)

	PlanetMath: triangular matrix
		A triangular matrix is a matrix that is an upper triangular matrix or lower triangular matrix.
		The eigenvalues of a triangular matrix are the diagonal elements.
		This is version 8 of triangular matrix, born on 2002-01-16, modified 2006-10-21.
	planetmath.org /encyclopedia/TriangularMatrix.html (306 words)

	Triangular matrix - Wikipedia, the free encyclopedia
		A triangular matrix with zero entries on the main diagonal is strictly upper or lower triangular.
		A matrix which is simultaneously upper and lower triangular is diagonal.
		The transpose of an upper triangular matrix is a lower triangular matrix and vice versa.
	en.wikipedia.org /wiki/Triangular_matrix (502 words)

	Triangular Matrix
		Assigns a matrix expression to the triangular matrix.
		Subtracts the matrix expression from the triangular matrix.
		Subtracts the matrix expression from the triangular adaptor.
	www.boost.org /libs/numeric/ublas/doc/triangular.htm (378 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - STRI, DTRI, STPI, ...
		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.
		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.
		Because matrix A is unit triangular, the diagonal elements are not referenced.
	www.nersc.gov /vendor_docs/ibm/essl/essl157.html (868 words)

	Linear Algebra (Math 2318) - Systems of Equations and Matrices - Special Matrices
		In other words, in a diagonal matrix is any matrix in which the only potentially non-zero entries are one the main diagonal. Any entry off the main diagonal must be zero and note that it is possible to have one or more of the main diagonal entries be zero.
		Note as well that a diagonal matrix can be thought of as both an upper triangular matrix and a lower triangular matrix.
		The final kind of matrix that we want to look at in this section is that of a symmetric matrix. In fact we’ve already seen these in a previous section we just didn’t have the space to investigate them in more detail in that section so we’re going to do it here.
	tutorial.math.lamar.edu /AllBrowsers/2318/SpecialMatrices.asp (1082 words)

	NMath Matrix User's Guide - 2.1 Lower Triangular Matrices
		A lower triangular matrix is a square matrix with all elements above the main diagonal equal to zero.
		Lower triangular matrices often arise at an intermediate stage in solving systems of equations and inverting matrices.
		NMath Matrix provides lower triangular matrix classes for four datatypes: single- and double-precision floating point numbers, and single- and double-precision complex numbers.
	www.centerspace.net /doc/NMath/Matrix/user/matrixtypes2.html (146 words)

	Guide and Reference
		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.
		To calculate the location of each element of the triangular matrix in the array, use the technique described in "Upper-Packed Storage Mode".
	www.navo.hpc.mil /usersupport/IBM/ESSL/essl139.html (580 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)

	Special Matrices: Triangular, Symmetric, Diagonal
		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)

	luinc (MATLAB Function Reference) (Site not responding. Last check: 2007-11-03)
		produces a unit lower triangular matrix, an upper triangular matrix, and a permutation matrix.
		returns the strict lower triangular part of the factor and the upper triangular factor embedded within the same matrix.
		The only exceptions to this dropping rule are the diagonal entries of the upper triangular factor, which are preserved to avoid a singular factor.
	www.ece.umr.edu /computing/unix/software/matlab/techdoc/ref/luinc.html (608 words)

	Matrix Factorization and Matrix Norms
		Several different matrix norms are discussed in the course reader and one in particular, the spectral norm, is discussed in the lecture material.
		We denote the spectral norm of the matrix by
		Several properties that hold for the spectral norm of a matrix were given in the lecture and you were asked to prove them as an un-graded homework problem.
	www.ee.ucla.edu /~brien/Rec5_MatrixFactorizationAndNorms.htm (718 words)

	Operations on a Triangular Matrix (Site not responding. Last check: 2007-11-03)
		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)

	Matrix Decomposition Methods
		After successful decomposition, the original matrix A is replaced by the orthogonal matrix Q, and the upper triangular matrix R is returned in r.
		Here r is the upper triangular matrix returned by the qrd method, v is the right hand side vector, and s is the reference to a valarray class object to receive the solution vector.
		The matrix is inverted in-place using minimum memory, i.e., the original matrix is replaced by its inverse.
	www.techsoftpl.com /matrix/doc/matdcmp.htm (995 words)

	CSI 801 Concepts Assignment Four Results (Site not responding. Last check: 2007-11-03)
		Recursively expand the determinant using the last column of the lower- triangular matrix.
		The inverse of Li of (6.1.13) is the same matrix with the signs of the off-diagonal elements changed.
		R is upper triangular, so R' is lower triangular, and R~ is upper triangular; but R~ = R' by orthogonality.
	www.science.gmu.edu /~jsteidel/801-ca4/ca4results.html (1347 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		CTRSM 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' or op(A) = conjg(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.
		UPLO = 'L' or 'l' A is a lower triangular matrix.
	www.ibiblio.org /gferg/ldp/man/manl/ctrsm.l.html (525 words)

	CenterSpace API Documentation - CenterSpace.NMath.Matrix
		A banded matrix is a matrix that has all its non-zero entries near the diagonal.
		A tridiagonal matrix is a matrix which has all its non-zero entries on the main diagonal, the super diagonal, and the subdiagonal.
		An upper triangular matrix is a square matrix with all elements below the main diagonal equal to zero.
	www.centerspace.net /doc/NMath/Matrix/ref (1869 words)

	ztrti2 - compute the inverse of a complex upper or lower triangular matrix (Site not responding. Last check: 2007-11-03)
		Specifies whether the matrix A is upper or lower triangular.
		Specifies whether or not the matrix A is unit triangular.
		On entry, the triangular matrix A. If UPLO = 'U', the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced.
	docs.sun.com /source/816-2461/ztrti2.html (307 words)

	Gauss Transformation and LU-Factorization (Site not responding. Last check: 2007-11-03)
		is upper triangular in columns 1 to k-1, and for the calculation of the elements of the Gauss matrix M
		It turns out that for a regular matrix it is possible after an appropriate interchange of matrix rows to find the LU factotization.
		Multiplying by the permutation matrix P on the left,, we obtain a new matrix, where the rows of initial the matrix are reordered exactly in the same way as the rows of the identity I are reordered for getteing P.
	www.cs.ut.ee /~toomas_l/linalg/lin2/node4.html (456 words)

	CTRMM
		CTRMM - perform one of the matrix-matrix operations B := alphaop(A)B, or B := alphaBop(A) where alpha is a scalar, B is an m by n matrix, 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’ or op(A) = conjg(A’)
		UPLO = ’U’ or ’u’ A is an upper triangular matrix.
		Before entry with UPLO = ’U’ or ’u’, the leading k by k upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced.
	www.math.ucla.edu /computing/docindex/blasman-man-24.html (494 words)

	Nilpotent matrix - Wikipedia, the free encyclopedia
		In mathematics, a nilpotent matrix is an n×n square matrix M such that
		There is a classification theorem showing that this is typical: a nilpotent matrix is similar to a block matrix, with diagonal square blocks generalising this type, and other blocks zero.
		Every strictly upper triangular matrix or strictly lower triangular matrix is nilpotent.
	en.wikipedia.org /wiki/Nilpotent_matrix (262 words)

	One iteration of the new method applied to upper triangular semiseparable matrices.
		In this section we describe one iteration of the proposed algorithm for computing the singular values of an upper triangular semiseparable matrix.
		As seen in Section 8.3.4, the application of this Givens transformation to the right of the lower triangular semiseparable matrix creates a bulge.
		The latter matrix is swapped back to upper triangular form.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node100.html (444 words)

	Chasing the bulge
		The matrix (8.8) differs from a lower triangular semiseparable matrix, because there is a bulge in position
		In order to retrieve the lower triangular semiseparable structure an algorithm is presented in this section.
		is lower triangular, the semiseparable structure is lost in its first two rows.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node99.html (312 words)

	[No title]
		uplo char specifies whether the matrix a is an upper or lower triangular matrix as follows: uplo = 'U' or 'u' : A is an upper triangular matrix.
		uplo = 'L' or 'l' : A is a lower triangular matrix.
		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.sandia.gov /ASCI/Red/usage/paragon/man/man3/_trmm.3.html (392 words)

	lu :: Functions (MATLAB Function Reference)
		is a square matrix with a mostly symmetric structure and mostly nonzero diagonal, UMFPACK uses a symmetric pivoting strategy.
		Upper triangular matrix that is a factor of
		Generate a 60-by-60 sparse adjacency matrix of the connectivity graph of the Buckminster-Fuller geodesic dome.
	www.mathworks.com /access/helpdesk/help/techdoc/ref/lu.shtml (783 words)

	Cholesky
		a method that obtains the Cholesky factorization RR´, where R is a lower triangular matrix, of a symmetric positive definite matrix A. a method to invert a symmetric positive definite matrix.
		This method factors the n by n symmetric positive definite matrix A as RR´ where R is a lower triangular matrix.
		The method proceeds by first factoring A as RR´ where R is a lower triangular matrix.
	www1.fpl.fs.fed.us /linear_algebra/Cholesky.html (610 words)

	Matlab manual page: 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.
		is a permutation of a lower triangular matrix that has 1s on the permuted diagonal, and that
		This quantity is produced by several of the LINPACK subroutines as an estimate of the reciprocal condition number of the input matrix.
	www.utexas.edu /its/rc/answers/math/matlab/manual/lu.html (264 words)

	Linear Algebra (Math 2318) - Systems of Equations and Matrices - LU-Decompositions (Site not responding. Last check: 2007-11-03)
		A into a product of a lower triangular matrix, L, and an upper triangular matrix, U. Such a factorization can be used to solve systems of equations as we’ll see in the next section when we revisit that topic.
		recall that the inverse matrix can be found by applying the inverse operation to the identity matrix.
		This is clearly an LU-Decomposition since the first matrix is lower triangular and the second is upper triangular and you should verify that upon multiplying they do in fact give the shown matrix.
	tutorial.math.lamar.edu /AllBrowsers/2318/LUDecomposition.asp (1349 words)

	Solution of TriangularSystems (Site not responding. Last check: 2007-11-03)
		the inverse of an upper (lower) triangular matrix is upper (lower) triangular;
		the inverse of an unit upper (lower) triangular matrix is unit upper (lower) triangular;
		the product of two unit upper (lower) triangular matrices is upper (lower) triangular.
	www.cs.ut.ee /~toomas_l/linalg/lin2/node3.html (153 words)

	1.2 Solution by a direct method (Site not responding. Last check: 2007-11-03)
		, where L is the lower triangular matrix with unit diagonal, and D is the diagonal matrix.
		To start with, we assume that matrix A can be stored entirely in main memory, so that the storage of the matrix and the right-hand-side vector can be described exactly.
		In practice, matrix B has the form B(NDSM,NTDL) where NDSM designates, for example, the number of load cases of RHS and NTDL is the number of degrees of freedom.
	www.math.psu.edu /local_doc/modulef/GB/Guide5-14/node7.html (1379 words)

	Cholesky.java
		On exit, the lower triangle of A has been replaced by R. @param a[ ][ ] The positive definite matrix to factor.
		The method assumes that at least the lower* triangle of a[ ][ ] is filled on entry.
		On exit the lower* triangle of a[ ][ ] has been replaced by R such that RR´ = A. @param n The order of the matrix* a[ ][ ].
	www1.fpl.fs.fed.us /Cholesky.html (843 words)

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