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Topic: Schur decomposition

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Schur triangulation

Schur complement

Issai Schur

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LU factorization

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	Schur decomposition - Wikipedia, the free encyclopedia
		In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation (named after Issai Schur) is an important matrix decomposition.
		Furthermore, if A is positive definite, the Schur decomposition of A is the same as the singular value decomposition of the matrix.
		Some algorithms in numerical linear algebra require a means of computing a Schur decomposition for a matrix.
	en.wikipedia.org /wiki/Schur_decomposition (338 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.
		In numerical analysis for example different decompositions are used to implement efficient matrix algorithms.
	en.wikipedia.org /wiki/Matrix_decomposition (131 words)

	[No title]
		The Schur decomposition, as I understand it, can be applied to any square matrix and produces a decomposition A = U * R * U', where U is unitary and R is upper triangular.
		A Schur decomposition is only appropriate for the LIMITED subclass of 'normal' matrices, which are known to be 'unitarily diagonalizable' (with all distinct -or- multiple eigenvalues).
		Besides normal matrices, the schur decomposition is NOT APPROPRIATE in the general case.
	www.octave.org /mailing-lists/bug-octave/2001/67 (707 words)

	QR decomposition -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The QR decomposition is often used to solve the (Click link for more info and facts about linear least squares) linear least squares problem.
		The QR decomposition is also the basis for a particular (Click link for more info and facts about eigenvalue algorithm) eigenvalue algorithm, the (Click link for more info and facts about QR algorithm) QR algorithm.
		Thus, we have, and the QR decomposition is.
	www.absoluteastronomy.com /encyclopedia/Q/QR/QR_decomposition1.htm (996 words)

	Advanced Topic: Matrix Decompositions
		QR decomposition is often used in solving least-squares fitting problems, and is typically faster than singular value decomposition.
		Schur decomposition is often used in evaluating functions of 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)

	Mathematica Documentation: Schur Decomposition
		However, the Schur decomposition can be found and the matrix transformed to an upper triangular form.
		Note that if the matrix is complex the definition of the Schur decomposition uses the conjugate transpose and returns an upper triangular result.
		With this new matrix the definition of the Schur decomposition can be seen as follows.
	documents.wolfram.com /mathematica/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/LinearAlgebraInMathematica/MatrixComputations/MatrixDecompositions/LinearAlgebra3.4.5.html (494 words)

	MUG: Schur Decomposition (Linear Algebra) (14.5.97) (Site not responding. Last check: 2007-10-08)
		The purpose of the Schur form is to give you everything that the Jordan form gives you, but in a numerically stable (and well-posed) way.
		So for a quick computation of one particular Schur form for one example (because time is too precious to be spent building general-purpose code just for one example) the proposed piece of code is ok, though nearly pessimal; under no circumstances should it ever be used more than once.
		This series of requests for the Schur factorization should be motivation for WMI to make the internal Schur factorization of evalf(Eigenvals(A)) (which must be there) available for general use.
	www.math.rwth-aachen.de /mapleAnswers/html/340.html (585 words)

	Issai Schur - Enpsychlopedia (Site not responding. Last check: 2007-10-08)
		Nevertheless he was dismissed from his chair in 1935 and, at the instigation of Bieberbach (who had previously sympathised with Schur regarding his treatment at the hands of the Nazis), he was forced to resign from the Prussian Academy in 1938.
		Schur eventually emigrated to Palestine in 1939, and had to endure the humiliation of living out his final years in poverty.
		He is perhaps best known today for his result on the existence of the Schur decomposition.
	www.grohol.com /psypsych/Schur (304 words)

	GNU Octave - Linear Algebra (Site not responding. Last check: 2007-10-08)
		The eigenvalues (and eigenvectors) of a matrix are computed in a several step process which begins with a Hessenberg decomposition, followed by a Schur decomposition, from which the eigenvalues are apparent.
		The Hessenberg decomposition is usually used as the first step in an eigenvalue computation, but has other applications as well (see Golub, Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979.
		The eigenvalues of are the diagonal elements of If the matrix is real, then the real Schur decomposition is computed, in which the matrix is orthogonal and is block upper triangular with blocks of size at most blocks along the diagonal.
	sunsite.univie.ac.at /textbooks/octave/octave_18.html (933 words)

	S-PLUS help (Site not responding. Last check: 2007-10-08)
		Schur Decomposition of a Matrix --- Generic Function
		Computes the Schur decomposition (including eigenvalues) of a square matrix.
		The Schur form is used most often for computing non-symmetric eigenvalue decompositions, and for computing functions of matrices such as matrix exponentials.
	www.uni-muenster.de /ZIV/Mitarbeiter/BennoSueselbeck/s-html/helpfiles/schur.html (139 words)

	Issai Schur (Site not responding. Last check: 2007-10-08)
		Issai Schur (January 10, 1875 in Mogilyov - January10, 1941 in Tel Aviv) was a mathematician who worked in Germany for most of his life.
		An ancillary resultwas the existence of the Schur decomposition which nowadayshe is best known for.
		See also: Schur's lemma, Schur indicator, Schur index, Schur complement.
	www.therfcc.org /issai-schur-239453.html (162 words)

	Eigenvalues, Eigenvectors and Generalized Schur Decomposition
		Another basic task is to compute the generalized Schur decomposition of the pair (A,B).
		Deflating subspaces are a generalization of invariant subspaces: the first k columns of Z span a right deflating subspace mapped by both A and B into a left deflating subspace spanned by the first k columns of Q.
		This decomposition is performed by the subroutine xGGHRD, which computes H and R, and optionally U and/or V.
	www.netlib.org /lapack/lug/node56.html (798 words)

	SciPy Scientific Tools for Python - module scipy.linalg.decomp (Site not responding. Last check: 2007-10-08)
		p,l,u -- LU decomposition matrices of a [permute_l=0]
		pl,u -- LU decomposition matrices of a [permute_l=1]
		QR decomposition of an M x N matrix a.
	www.scipy.org /documentation/apidocs/scipy/scipy.linalg.decomp.html (510 words)

	PlanetMath: corollary of Schur decomposition
		By Schur decompostion we know that there exists a
		"corollary of Schur decomposition" is owned by Daume.
		This is version 3 of corollary of Schur decomposition, born on 2003-06-29, modified 2003-12-13.
	planetmath.org /encyclopedia/CorollaryOfSchurDecomposition.html (146 words)

	Structure of the Eigenvalue Problem
		Schur's result states that every square matrix is unitarily similar to an upper triangular matrix.
		The fundamental structure of Hermitian and normal matrices is easily derived from the Schur decomposition.
		The iteration is continued until the subdiagonal elements of converge to zero, i.e.
	www.caam.rice.edu /software/ARPACK/UG/node46.html (1176 words)

	[No title]
		ZTGEXC - reorder the generalized Schur decomposition of a complex matrix pair (A,B), using an unitary equivalence transformation (A, B) := Q * (A, B) * Z', so that the diagonal block of (A, B) with row index IFST is moved to row ILST
		ZTGEXC reorders the generalized Schur decomposition of a complex matrix pair (A,B), using an unitary equivalence transformation (A, B) := Q * (A, B) * Z', so that the diagonal block of (A, B) with row index IFST is moved to row ILST.
		Optionally, the matrices Q and Z of generalized Schur vec tors are updated.
	www.ibiblio.org /gferg/ldp/man/manl/ztgexc.l.html (594 words)

	Generalized Nonsymmetric Eigenproblems (GNEP) (Site not responding. Last check: 2007-10-08)
		The generalized nonsymmetric eigenvalue problem can be solved via the generalized Schur decomposition of the matrix pair (A, B), defined in the real case as
		As for the standard nonsymmetric eigenproblem, two pairs of drivers are provided, one pair focusing on the generalized Schur decomposition, and the other pair on the eigenvalues and eigenvectors as shown in Table 2.6:
		To save space in Table 2.6, the word ``generalized'' is omitted before Schur decomposition, eigenvalues/vectors and singular values/vectors.
	www.go.dlr.de:8081 /pdinfo_dv/lapack_lug_dlr/node35.html (564 words)

	Deflating Subspaces and Condition Numbers
		The generalized Schur form depends on the order of the eigenvalues on the diagonal of (S,T) and this may optionally be chosen by the user.
		It may be used to choose the order in which eigenvalues appear in the generalized Schur form.
		These are the same as the condition numbers of the eigenvalues and eigenvectors of the original matrix pair (A,B), from which (S,T) is derived.
	www.netlib.org /lapack/lug/node58.html (398 words)

	ACM TOMS: Bibliographic record for `Demmel:1993:GSDb' (Site not responding. Last check: 2007-10-08)
		The decomposition is a generalization of the Schur canonical form of $A - \lambda I$ to matrix pencils and reveals the Kronecker structure of a singular pencil.
		The second part of this two-part paper describes the computed generalized Schur decomposition in more detail and the software, and presents applications and an example of its use.
		Background theory and algorithms for the decomposition and its error bounds are presented in Part I of this paper.", keywords = "algorithms; geig; generalized Schur decomposition; matrix pencil; nla; reliability; theory", subject = "{\bf G.1.3}: Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Eigenvalues.
	math.nist.gov /toms/cgi-bin/TOMSbibget?Demmel:1993:GSDb (168 words)

	HAPACK FORTRAN 77 routines
		Test routine: tdshes.f tdshes.f plus dependencies Test data: dshes.dat dshes.res file dshevx.f dshevx.f plus dependencies Computes the eigenvalues and eigenvectors of a skew-Hamiltonian matrix, with preliminary balancing of the matrix, and computes reciprocal condition numbers for the eigenvalues and some eigenvectors.
		Test routine: tdgesqb.f tdgesqb.f plus dependencies Test data: dgesqb.dat dgesqb.res file dgesqr.f Symplectic QR decomposition of a general matrix.
		Test routine: tdhapvl.f tdhapvl.f plus dependencies Test data: dhapvl.dat dhapvl.res file dhgpqr.f dhgpqr.f plus dependencies Periodic Schur decomposition of a product of two matrices.
	www.math.tu-berlin.de /~kressner/hapack/routines (832 words)

	The Schur Method
		First, it modifies the open-loop eigenvalue(s) contained in the uppermost diagonal block and the Schur form is then reordered to bring the next open-loop eigenvalue(s) to be reassigned in the next step on the top of the Schur form, and so on.
		The Schur method is the most expensive of the direct methods if it is used to assign all the poles.
		From the preceding discussion, however, it is the best-suited method to solve the partial pole assignment problem (see Section 5.2).
	documents.wolfram.com /applications/anm/PoleAssignment/5.1.3.html (209 words)

	7.4 Schur Decompositions (Site not responding. Last check: 2007-10-08)
		The Schur decomposition of a matrix is closely related, and often more useful, than the eigenvalues and eigenvectors.
		Like an eigenvalue decomposition, the Schur decomposition reveals the eigenvalues of a matrix and the invariant subspaces.
		A Schur decomposition object contains member functions for accessing the eigenvalues of the matrix A, and the components of the decomposition.
	www.roguewave.com /support/docs/hppdocs/lpkug/7-4.html (186 words)

	Issai Schur - Encyclopedia, History, Geography and Biography
		As a student of Ferdinand Georg Frobenius, he worked on group representations but also in combinatorics and even theoretical physics.
		See also: Schur's lemma, Schur multiplier, Schur indicator, Schur index, Schur complement, Schur's theorem.
		This page was last modified 00:14, 27 May 2005.
	www.arikah.net /encyclopedia/Issai_Schur (222 words)

	chseqr(l): compute eigenvalues of complex upper ... - Linux man page
		CHSEQR - compute the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T ZH, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur** vectors
		CHSEQR computes the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T ZH, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur** vectors.
		= 'N': no Schur vectors are computed; = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned; = 'V': Z must contain an unitary matrix Q on entry, and the product Q*Z is returned.
	www.die.net /doc/linux/man/manl/chseqr.l.html (513 words)

	Schur decomposition -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Schur decomposition -- Facts, Info, and Encyclopedia article
		If A is a square matrix over the (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers, then A can be decomposed as
		Furthermore, if A is (Click link for more info and facts about positive definite) positive definite, the Schur decomposition of A is the same as the (Click link for more info and facts about singular value decomposition) singular value decomposition of the matrix.
	www.absoluteastronomy.com /encyclopedia/s/sc/schur_decomposition.htm (157 words)

	S-PLUS help (Site not responding. Last check: 2007-10-08)
		Computes the Schur decomposition and eigenvalues of a square matrix.
		either a logical value or a logical vector of length 2 telling whether or not to compute reciprocal condition numbers for the eigenvalues and Schur vectors, respectively, and return them as attributes.
		A diagonal block associated with a particular eigenvalue will appear in the top left portion of the Schur decomposition if it is greater than or equal to
	www.uni-muenster.de /ZIV/Mitarbeiter/BennoSueselbeck/s-html/helpfiles/schur.Matrix.html (247 words)

	stgexc - reorder the generalized real Schur decomposition of a real matrix pair * Z',
		A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks.
		Optionally, the matrices Q and Z of generalized Schur vectors are updated.
		On entry, the matrix B in generalized real Schur canonical form (A,B).
	docs.sun.com /source/816-2461/stgexc.html (644 words)

	ctgexc - reorder the generalized Schur decomposition of a complex matrix pair with row index IFST is moved to row ILST
		ctgexc - reorder the generalized Schur decomposition of a complex matrix pair with row index IFST is moved to row ILST
		ctgexc - reorder the generalized Schur decomposition of a complex matrix pair (A,B), using an unitary equivalence transformation (A, B) := Q * (A, B) * Z', so that the diagonal block of (A, B) with row index IFST is moved to row ILST
		ctgexc reorders the generalized Schur decomposition of a complex matrix pair (A,B), using an unitary equivalence transformation (A, B) := Q * (A, B) * Z', so that the diagonal block of (A, B) with row index IFST is moved to row ILST.
	docs.sun.com /source/816-2461/ctgexc.html (639 words)

	Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition
		Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition: SIAM Journal on Matrix Analysis and Applications Vol.
		The canonical decomposition of higher-order tensors is a key tool in multilinear algebra.
		A first-order perturbation analysis of the simultaneous generalized Schur decomposition is carried out.
	epubs.siam.org /sam-bin/dbq/article/39786 (223 words)

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