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Topic: Jordan decomposition

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	Decomposition (Site not responding. Last check: 2007-10-29)
		Mathematical model decomposition structure may be treated also as bourbakis structures is generated on the set of decomposition (P- and F-) by original bourbakis structure of the object (model).
		Jordan representation of linear operator in finite-dimensional linear vector space over complex number field is maximal DP-decomposition of the operator on family of DP-simple (see lower) factor-object of the operator.
		Decomposition in such system is mean to realize hierarchy of value, which are achieved the system, to bring in correspondence with complexity of control tasks with resource for its solution and information technology inbuilt in the system.
	www.ccas.ru /Pavlovsky/Edecomp.htm (1298 words)

	Wikinfo \| Spectral theorem (Site not responding. Last check: 2007-10-29)
		In mathematics, the spectral theorem is an important decomposition theorem applying to normal operators in linear algebra and functional analysis.
		Since the class of normal operators includes a number of special kinds of operators such as symmetric operators or unitary operators or special kinds of matrices for example, a symmetric matrix, the spectral theorem can be applied in a wide range of situations.
		Jordan decomposition, an "algebraic" analogue to spectral decomposition.
	www.wikinfo.org /wiki.php?title=Spectral_theorem (1065 words)

	PlanetMath: Jordan decomposition
		This definition is easily shown to be independent of the chosen Hahn decomposition.
		Cross-references: negative, finite, measure, positive, clear, independent, Hahn decomposition, signed measure
		This is version 6 of Jordan decomposition, born on 2003-02-10, modified 2003-02-11.
	planetmath.org /encyclopedia/JordanDecomposition.html (91 words)

	Advanced Topic: Matrix Decompositions
		QR decomposition is often used in solving least-squares fitting problems, and is typically faster than singular value decomposition.
		Jordan decomposition is often used in evaluating functions of exact matrices.
		LU decomposition is convenient for preprocessing matrices that will appear repeatedly in collections of linear equations, as discussed in Section 3.7.8.
	documents.wolfram.com /v4/MainBook/3.7.10.html (439 words)

	Springer Online Reference Works (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-10-29)
		A Jordan decomposition is also the representation of a signed measure or a charge
		This extension of the concept of a Jordan decomposition allows one to introduce the concept of a Jordan decomposition in algebraic groups and algebras.
		The concept of a Jordan decomposition in algebraic groups and algebraic Lie algebras allows one to introduce the definitions of a semi-simple and a unipotent (nilpotent, respectively) element in an arbitrary affine algebraic group (algebraic Lie algebra, respectively).
	eom.springer.de.cob-web.org:8888 /J/j054310.htm (542 words)

	eFunda: Jordan Decomposition (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-10-29)
		Usually, the results of the decomposition are a diagonal matrix formed by eigenvalues and a matrix formed by eigenvectors.
		Thus, it is also known as the eignevalue/eigenvector decomposition when all eigenvalues are distinct.
		However, when repeated roots of nonzero eigenvalues occur, the Jordan canonical form is used to "diagonalize" the undiagonalizable matrix.
	www.efunda.com.cob-web.org:8888 /webM/matrix/jordandecomp1.cfm (131 words)

	Ken Jordan's Theoretical Chemistry Group
		Jordan' "Infrared Spectrum and Structural Assignment of the Water Trimer Anion", J.
		Sorescu and K. Jordan, "Theoretical Study of the Adsorption of Acetylene on the Si(001) Surface", J.
		Nachtigall, K. Jordan, and C. Sosa, "Theoretical Study of the Mechanism of Recombinative Hydrogen Desorption from the Monohydride Phase of Si(100): The Role of Defect Migration", J. Chem.
	www.pitt.edu /~jordan/references.html (2764 words)

	Matrix Decompositions
		The LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation.
		The first element is a combination of upper and lower triangular matrices, the second element is a vector specifying rows used for pivoting (a permutation vector which is equivalent to the permutation matrix), and the third element is an estimate of the condition number.
		This is often referred to as the thin QR decomposition, see for example Golub and van Loan.
	documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/4.5.html (2005 words)

	Self-adjoint Eigenproblems
		The spectral information one can request from a singular value decomposition is similar, except that there are left and right singular vectors and singular subspaces, corresponding to eigenvectors and invariant subspaces.
		Jordan form is usually called Weierstrass form for regular pencils, and Kronecker form for singular pencils.
		In addition to these decompositions, the user may request condition numbers for any of the computed quantities (eigenvalues, means of eigenvalue clusters, eigenvectors, invariant/deflating/reducing subspaces) [13][38][27].
	www.netlib.org /utk/papers/etemplates/node17.html (494 words)

	Linear Algebra, MTH 513
		Topics covered from: eigenvalues, eigenvectors, eigensystems, unitary matrices and transformations, Shur decomposition, the QR decomposition, the singular-value decomposition, the Jordan form, Hermitian matrices, and definite quadratic forms.
		We will study several such decompositions, which are not only useful in obtaining knowledge of the structure of the eigensystems but for other applications as well.
		The Jordan form is useful in analyzing defective matrices, those which are not diagonalizable.
	hypatia.math.uri.edu /~eaton/mth513.f01.htm (1120 words)

	Schur's Decomposition
		Consequently, we have obtained the Schur decomposition of the matrix A.
		Since a triangular matrix is normal only if it is a diagonal matrix, then it has been proved that a unitary matrix is similar to a diagonal matrix.
		Jordan block,and the matrix J is called the Jordan normal form of the matrix A.
	www.cs.ut.ee /~toomas_l/linalg/lin1/node17.html (645 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		From: rusin@math.niu.edu (Dave Rusin) Newsgroups: sci.math.research Subject: Re: Continuity of the Jordan Decomposition Map Date: 27 Feb 1998 23:09:27 GMT In article
		wrote: >Consider the Jordan decomposition of an nXn matrix M = Q D Q^{-1}.
		On the other hand, as M is continously deformed, it can easily change from diagonalizable to not (e.g.
	www.math.niu.edu /~rusin/known-math/98/jordan_decomp (158 words)

	Linear Algebra - Preliminary lectures
		Cholesky decomposition is based on the following extension of this property.
		Similarly to the LU decomposition, the Gram-Schmidt orthogonalisation technique is used for calculating
		The computation of the singular value decomposition is therefore simply done by calculating the eigenvalues and eigenvectors of
	vision.unige.ch /~marchand/teaching/linalg/linalg.html (1819 words)

	Matrix decomposition - Wikipedia, the free encyclopedia
		In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form.
		There are several different decompositions of a given matrix and the decomposition used depends on the problem we want to solve as well as the matrix to be factorized.
		In numerical analysis for example different decompositions are used to implement efficient matrix algorithms.
	en.wikipedia.org /wiki/Matrix_decomposition (158 words)

	Algorithm of Filipov
		If n=1, then the Jordan block coincides with the given matrix and formula (9) is true.
		Let us suppose that the Jordan form of the matrix A is found by applying the Jordan block construction formula (9) if the order of the matrix A is smaller than n.
		Since the matrix X in the Jordan decomposition of the matrix A is not uniquely defined, then for many problems it is of interest to choose the matrix X so that the conditional number k(X) were the least.
	www.cs.ut.ee /~toomas_l/linalg/lin2/node21.html (242 words)

	PlanetMath: Jordan-Hölder decomposition theorem
		satisfying the properties above is called a Jordan-Hölder decomposition of
		Cross-references: properties, ordering, multiset, decompositions, simple group, quotient group, normal, filtration, finite group
		This is version 5 of Jordan-Hölder decomposition theorem, born on 2002-01-05, modified 2004-06-23.
	planetmath.org /encyclopedia/JordanHolderDecompositionTheorem.html (60 words)

	Huajun's Homepage (Site not responding. Last check: 2007-10-29)
		Given a representation of a real or complex reductive Lie group G on V, a parabolic subgroup P imposes certain partial order (or grading) on V. One can describe the parabolic subgroup orbits and invariants on V by studying the structure of G in accordance with the partial order.
		They are relative to the Bruhat decomposition, Iwasawa decomposition, and some topics in representation theory.
		Some classical decompositions are Jordan decomposition, Cartan decomposition, Polar decomposition, Bruhat decomposition, Iwasawa decomposition, and Levi decomposition.
	www.auburn.edu /~huanghu (231 words)

	Linear Algebra, MTH 513 <br>Syllabus Fall 2000 <br> Mon. Wed. 4:30 - 5:45, Rm 106, Tyler Hall
		Topics covered: eigenvalues, eigenvectors, eigensystems, unitary matrices and transformations, Shur decomposition, the QR decomposition, the singular-value decomposition, the Jordan form, and definite quadratic forms.
		A basic knowlege of linear algebra such as is covered in an undergraduate course is required as background for this course.
		We will study several such decompositions, which are not only useful in obtaining knowlege of of the structure of the eigensystems but for other applications as well.
	www.math.uri.edu /~eaton/mth513f00.html (946 words)

	The Mathematica Book Online: Mathematica Reference Guide \| Some Notes on Internal Implementation
		Integer powers are found by a left-right binary decomposition algorithm.
		Fourier uses the FFT algorithm with decomposition of the length into prime factors.
		Sparse arrays are stored internally using compressed sparse row formats, generalized for tensors of arbitrary rank.
	documents.wolfram.com /mathematica/book/section-A.9.4 (1362 words)

	Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia
		The fact that Jordan blocks in general are not diagonal but nilpotent is directly related to the distinction between eigenvectors and generalized eigenvectors.
		The decomposition theorem is a version of the spectral theorem in the particular case of matrices.
		The number and the sizes of the Jordan blocks are dictated by the geometric and algebraic multiplicities of the eigenvalues.
	en.wikipedia.org /wiki/Eigenvector (5429 words)

	Citebase - A short constructive proof of Jordan's decomposition theorem
		A short constructive proof of Jordan's decomposition theorem
		Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary.
		In fact, all we need is the theorem on the dimensions of rang and kernel and the existence of eigenvalues of a linear transformation on a nontrivial finite dimensional complex vector space.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0312041 (132 words)

	Eigenvalue order (Site not responding. Last check: 2007-10-29)
		On the other hand, when working a discrete-time optimal control problem, the eigenvalues of the corresponding symplectic matrix need to be ordered by magnitude.
		Of course, in both these cases, one should not compute these values by the Jordan (eig) decomposition, but rather through the schur() command, which is a first step in the eig computation.
		It's trivial to write an m-file for the Jordan decomposition obtained with eig().
	www.cae.wisc.edu /pipermail/bug-octave/2006-May/000305.html (286 words)

	Novosibirsk SU contest 1
		, where J is of the form described above, is called a Jordan decomposition of A.
		The Jordan decomposition of a matrix may fail to be unique.
		It can be proved that exp A is independent of the Jordan decomposition of A used.
	acm.uva.es /p/v107/10753.html (265 words)

	Egwald Mathematics - Linear Algebra: Matrices and Matrix Decomposition
		Otherwise, the Jordan Normal Form augmented with generalized eigenvectors is used to decompose the matrix.
		The objective of Gaussian Decomposition is to write a matrix A as the product of a lower triangular matrix L and an upper triangular matrix U. The Gaussian method is similar to the Gauss-Jordan method for computing a matrix's inverse.
		With the P * A = L * U decomposition the determinant of A, det(A), is the product of the diagonal elements of the matrix U, times +1 or -1 if an even or an odd number of row switches were used in the decomposition.
	www.egwald.com /linearalgebra/matrices.php (3976 words)

	jordan decomposition and generalized eigenvectors (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-10-29)
		jordan decomposition and generalized eigenvectors (via CobWeb/3.1 planetlab2.tamu.edu)
		anyway, i am using an algorithm to perform a jordan decomposition taken from
		The Jordan form is known to be notoriously instable.
	sci4um.com.cob-web.org:8888 /about33970.html (1887 words)

	Linear algebra
		Cholesky(mat) computes the Cholesky decomposition of a square symmetric positive-definite matrix mat and returns a lower-triangular matrix L such that mat=L×L
		Gram-Schmidt orthonormalization is known to be numerically unstable (see “Matrix Computations” by Golub and van Loan), but should be fine for symbolic manipulation.
		Note: Jordan decomposition is often used for evaluating symbolic functions of matrices, such as matrix exponentials.
	www.ibiblio.org /technicalc/packages/mathtools/linearalgebra.htm (1176 words)

	Table of contents for Library of Congress control number 2003055189 (Site not responding. Last check: 2007-10-29)
		Jordan decomposition as a Morita equivalence, the main reductions 11.
		Jordan decomposition as a Morita equivalence, sheaves 12.
		Decomposition numbers and q-Schur algebras, linear primes Part V. Unipotent Blocks and Twisted Induction: 21.
	www.loc.gov /catdir/toc/cam032/2003055189.html (188 words)

	[No title]
		Four MATLAB programs for the Jordan matrix decomposition of complex floating point matrices: 1) jordan.m top level program.
		3) nulld.m robust null space sizer 4) jordlook.m diagnostic program for tough null spaces function [M,D] = jordan(a) %Jordan Compute the Jordan decomposition of square matrix A % such that AM = MD. The elements of A may be real % or complex floating point numbers.
		% Z = NULL(A) is an orthonormal basis for the null space of A obtained % from the singular value decomposition.
	my.fit.edu /~gabdo/jordan.txt (321 words)

	15: Linear and multilinear algebra; matrix theory
		Do the parts of the Jordan Decomposition of a matrix vary continuously with the matrix?
		Some pointers on the computation of the Singular Value Decomposition of a matrix.
		Maple code to do QR decomposition of a matrix.
	www.math.niu.edu /~rusin/known-math/index/15-XX.html (1605 words)

	Lectures 12-13
		Theorem: Suppose L is a semisimple Lie algebra and f : L ~> gl(V) is a faithful representation.
		As H is abelian ad(H) is a family of commuting semeisimple elements in gl(L) hence we can simultaneously diagonalize ad(H) of obtain the decomposition L = (+)
		This sum is a direct sum of vector spaces and not a direct sum of ideals as discussed before.
	www.math.rutgers.edu /~nacin/Sahi12.html (1089 words)

	Real Analysis 2
		This course is meant to continue the study of analysis of real-valued functions of one or several variables, with an emphasis on Lebesgue measure and Lebesgue integration on the real line and R^n.
		L^p - spaces, definitions and examples; Minkowski's inequality, Holder's inequality; Arzela-Ascoli Theorem, Stone-Weierstrass Theorem, Signed measures, definitions and basic properties, Hahn decomposition Theorem, Jordan decompostion, mutually singular measures, Jordan decomposition Theorem, comparison of measures, absolutely continuous measures, Radon-Nikodym Theorem, Lebesgue decomposition theorem; product measures: Fubini's Theorem, Tonelli's Theorem, applications of Fubini Theorem to integral operators.
		We will use as a primary text the book "Real Analysis", by H.L. Royden, Third Edition, covering most of Chapters 6, parts of Chapters 7 and 9, and most of Chapters 11, and 12.
	spot.colorado.edu /~packer/6320.html (721 words)

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