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Topic: Upper triangular matrix

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Lower triangular matrix

Crout decomposition

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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.
	www.planetmath.org /encyclopedia/StrictUpperTriangularMatrix.html (102 words)

	NationMaster - Encyclopedia: Triangular matrix
		In mathematics, a nilpotent matrix is a square matrix that is nilpotent.
		The transpose of a upper triangular matrix is a lower triangular matrix and vice versa.
		The eigenvalues of a triangular matrix are the diagonal elements.
	www.nationmaster.com /encyclopedia/Triangular-matrix (1200 words)

	Egwald Mathematics - Linear Algebra: Matrices and Matrix Decomposition
		A matrix of dimension m by 1 is called a column vector; a matrix of dimension 1 by n is called a row vector.
		of a square matrix A of dimension n is a measure of the "distance" of A from the zero matrix, O, with properties similar to those of a vector norm.
		A permutation matrix P is formed by rearranging the columns of an identity matrix.
	www.egwald.com /linearalgebra/matrices.php (3984 words)

	Triangular matrix - Wikipedia, the free encyclopedia
		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: 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/TriangularMatrix.html (249 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.
	planetmath.org /encyclopedia/StrictUpperTriangularMatrix.html (93 words)

	Triangular matrix -- Facts, Info, and Encyclopedia article (Site not responding. Last check: )
		A triangular matrix with zero entries on the (The diagonal of a square matrix running from the upper left entry to the lower right entry) main diagonal is strictly upper or lower triangular.
		A matrix which is simultaneously upper and lower triangular is (An oblique line of squares of the same color on a checkerboard) diagonal.
		The (A matrix formed by interchanging the rows and columns of a given matrix) transpose of a upper triangular matrix is a lower triangular matrix and vice versa.
	www.absoluteastronomy.com /encyclopedia/t/tr/triangular_matrix.htm (562 words)

	Guide and Reference
		When using upper- or lower-triangular storage, the unreferenced elements in the lower and upper triangular part, respectively, are assumed to be zero.
		Because matrix A is unit triangular, the diagonal elements are not referenced.
		Matrix A is a complex 4 by 4 upper unit triangular matrix, stored in upper-triangular storage mode.
	www.umiacs.umd.edu /~jhu/DOCS/SP/docs/essl/essl339.html (1108 words)

	Matrix Tutorial 1: Stochastic Matrices
		The sum of the principal is the trace of the matrix.
		Accordingly, the rank of a square matrix is equal to the number of nonzero rows in its upper-triangular matrix or the number of nonzero columns in its equivalent lower-triangular matrix, whichever of these two number is smaller.
		In general, a matrix whose sum of all row elements (or column elements) equals 1 is called a stochastic matrix.
	www.miislita.com /information-retrieval-tutorial/matrix-tutorial-1-stochastic-matrices.html (2152 words)

	Guide and Reference (Site not responding. Last check: )
		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)

	lower triangular matrix multiplication (Site not responding. Last check: )
		is in lower triangular portion of array A, and the diagonal of...
		is lower triangular with ones on the diagonal, is...
		Toyota Matrix is a 5-door styled crossover-over utility vehicle aimed for young buyers in Standard, XR and XRS trims.
	matrix1.myhairsource.com /lower-triangular-matrix-multiplication.html (322 words)

	Guide and Reference
		There are two types of triangular band matrices: upper triangular band matrix and lower triangular band matrix.
		A triangular band matrix can also be a unit triangular band matrix if all the diagonal elements have a value of 1.
		Only the band elements of the upper triangular part of an upper triangular band matrix, including the main diagonal, are stored for upper-triangular-band-packed storage mode.
	www.umiacs.umd.edu /~jhu/DOCS/SP/docs/essl/essl143.html (590 words)

	psi^3 as an upper triangular matrix, by Jonathan Barker and Victor Snaith (Site not responding. Last check: )
		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)

	Matrix Notation / Types of Matrices
		For instance, a matrix like this one, with all-zero entries below the top-left-to-lower-right diagonal ("the diagonal") is called "upper triangular".
		Note that triangular matrices are square, that diagonals are triangular and therefore are square, and that identities are diagonals and therefore are triangular and square.
		upper triangular matrix, but it is not diagonal.
	www.purplemath.com /modules/matrices2.htm (435 words)

	Crout matrix decomposition - Wikipedia, the free encyclopedia
		In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix (P).
		The Crout matrix decomposition algorithm differs slightly from the Doolittle method.
		Doolittle's method returns a unit lower triangular matrix and a upper triangular matrix where the Crout method returns a lower triangular matrix and a unit upper triangular matrix.
	en.wikipedia.org /wiki/Crout_matrix_decomposition (142 words)

	lufact.html
		You notice you can pull a diagonal matrix D off to the left of the upper triangular factor U so that what is left has 1's down the diagonal.
		The upper triangular factor can itself be factored into a diagonal times a unit upper triangular matrix whose transpose is the lower unit triangular factor.
		If A is a nonsingular symmetric matrix which factors into LU, then U is the product of a diagaonal matrix D with an upper unit triangular matrix U1 whose transpose is L. One big use of this theorem is to save storage.
	www.msc.uky.edu /carl/ma322/spr2001/html/lufact1.html (863 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)

	COMPORT Call
		If b is not specified, q is the m ×m orthogonal matrix Q that is the product of the min(m,n) separate Householder transformations.
		If b is specified, q is the m ×p matrix Q' B that has the transposed Householder transformations Q' applied on the p columns of the argument matrix B.
		specifies an optional m ×p matrix B that is to be left multiplied by the transposed m ×m matrix Q'.
	www.asu.edu /it/fyi/dst/helpdocs/statistics/sas/sasdoc/sashtml/iml/chap17/sect42.htm (425 words)

	[No title] (Site not responding. Last check: )
		The upper trapezoidal matrix A is fac tored as A = (R 0) * Z, where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix.
		M (input) INTEGER The number of rows of the matrix A. N (input) INTEGER The number of columns of the matrix A. A (input/output) REAL array, dimension (LDA,N) On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized.
		The elements of R are returned in the upper triangular part of A. Z is given by Z = Z(1) * Z(2) *...
	www.ibiblio.org /gferg/ldp/man/manl/stzrzf.l.html (397 words)

	LU Decomposition
		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.
		If a matrix A can be decomposed into an LU representation, then A is equal to the product of the lower and upper triangular matrices.
	www.aspire.cs.uah.edu /textbook/LU.html (650 words)

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

	Structure of the Eigenvalue Problem
		If where is unitary and is upper triangular(the standard QR-factorization), then where is an upper triangular matrix with the eigenvalues on its diagonal.
		An upper Hessenberg matrix has zero elements below its main subdiagonal so it is almost upper triangular.
		is Hermitian, is a real symmetric tridiagonal matrix in which case all the elements are zero except for those on the main, sub and super diagonals.
	www.caam.rice.edu /software/ARPACK/UG/node46.html (1176 words)

	[No title]
		UPLO = 'L' or 'l' A is a lower triangular matrix.
		On entry, M specifies the number of rows of the matrix C. M must be at least zero.
		upper triangular part of C is not referenced.
	orion.math.iastate.edu /burkardt/f_src/dblas/dblas3.f90 (3649 words)

	Schur's Theorem
		Schur's theorem (biography) states that any matrix is unitarily similar to an upper triangular matrix.
		We move to an upper triangular as a matter of convention.
		The upper left entry is still s, and the rest of the top row is, well, who knows.
	www.mathreference.com /la-sim,schur.html (1435 words)

	HLAPack.DP (Site not responding. Last check: )
		DPBRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and banded, and provides error bounds and backward error estimates for the solution.
		DPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution.
		D is a diagonal matrix specified in the vector D, L is a unit bidiagonal matrix whose subdiagonal is specified in the vector E, and X and B are N by NRHS matrices.
	www.isi.edu /~hdaume/HBlas/docs/HLAPack.DP.html (1810 words)

	[No title] (Site not responding. Last check: )
		The upper trapezoidal matrix A is factored as A = (R 0) * Z, where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix.
		M (input) INTEGER The number of rows of the matrix A. N (input) INTEGER The number of columns of the matrix A. A (input/output) COMPLEX16 array, dimension (LDA,N) On entry, the leading M-by-N upper* trapezoidal part of the array A must contain the matrix to be factorized.
		On exit, the leading M-by-M upper triangular part of A contains the upper triangular matrix R, and elements M+1 to N of the first M rows of A, with the array TAU, represent the uni tary matrix Z as a product of M elementary reflec tors.
	www.ibiblio.org /gferg/ldp/man/manl/ztzrzf.l.html (397 words)

	[No title]
		# where l is a lower triangular matrix.
		# matrix (a), with a flop count of (n^3)/3.
		# n is the dimension of the matrix a.
	www.math.unl.edu /~ijeib/Programs/Numerical_Analysis/PROJ94.htm (1891 words)

	Gauss Transformation and LU-Factorization
		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)

	Ldu decomposition - LearnThis.Info Enclyclopedia (Site not responding. Last check: )
		The permutation matrix can possibly be equal to the unit matrix I. If it is, than LDU is a unique decomposition, that is, for any other decomposition L!=L, D!=D, and U!=U*.
		Therefore the inverse of a lower-triangular matrix must also be a lower-triangular matrix.
		If you multiply a lower-triangular matrix by a diagonal matrix, you still have a lower-triangular matrix.
	encyclopedia.learnthis.info /l/ld/ldu_decomposition.html (215 words)

	NMath Matrix User's Guide - 2.2 Upper Triangular Matrices
		An upper triangular matrix is a square matrix with all elements below the main diagonal equal to zero.
		Like lower triangular matrices, upper triangular matrices often arise at an intermediate stage in solving systems of equations and inverting matrices.
		NMath Matrix provides upper 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/matrixtypes3.html (150 words)

	dtgsen(l): reorder generalized real Schur ... - Linux man page
		A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks.
		On exit, Q has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B); The leading M columns of Q form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces).
		On exit, Z has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B); The leading M columns of Z form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces).
	www.die.net /doc/linux/man/manl/dtgsen.l.html (1684 words)

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