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Topic: Generalized eigenvector

	Generalized eigenvector - Wikipedia, the free encyclopedia
		In linear algebra, a generalized eigenvector of a matrix A is a nonzero vector v, which has associated with it an eigenvalue Î» having algebraic multiplicity k, satisfying
		Generalized eigenvectors can be used to determine the Jordan form.
		The usage of generalized eigenfunction differs from this; it is part of the theory of rigged Hilbert spaces, so that for a linear operator on a function space this may be something different.
	en.wikipedia.org /wiki/Generalized_eigenvector (113 words)

	Pellionisz (1985) Tensor Network Theory of the Metaorganization of Functional Geometries in the Central Nervous System
		The stored eigenvectors and eigenvalues can serve, respectively, (1) as a means for the genesis of a metric (in the form of its spectral representation) with the given eigenvectors and (2) as a means of comparing the eigenvalues that are implicit in the external body geometry and those of the internal metric.
		The general hypothesis of the geometrical interpretation of brain function hinges on the assumption that the relation between the brain and the external world is determined by the ability of the CNS to construct an internal model of the external world using an interactive relationship between sensory and motor expressions.
		A general tensorial interpretation of the CNS is based on the notion that the intrinsic natural frames of reference, in which neurons attribute ordered sets of activity-values (co- and contravariant vectors) to physical invariants of the external world, invoke multidimensional arithmetic manifolds.
	www.usa-siliconvalley.com /inst/pellionisz/85_metaorganization/85_metaorganization.html (9761 words)

	Chapter 3
		Eigenvectors and eigenvalues are introduced geometrically ("Where does the vector field point directly toward or directly away from the origin?").
		We derive the general solution for a 2x2 system with repeated eigenvalues by taking advantage of the fact that every vector is a generalized eigenvector.
		This approach leads to general solutions that are expressed in terms of their initial conditions, and it should be contrasted with the approach that was taken in Sections 3.2-3.4.
	math.bu.edu /odes/paul-inst-manual/sed/ch3.html (2823 words)

	cgegv
		A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B is singular.
		A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - w B) r = 0.
		On entry, the second of the pair of matrices whose generalized eigenvalues and (optionally) generalized eigenvectors are to be computed.
	docs.sun.com /source/806-7993/cgegv.html (676 words)

	stgevc - compute some or all of the right and/or left generalized eigenvectors of a pair of real upper triangular ...
		If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of (A,B), or the products ZX and/or QY, where Z and Q are input orthogonal matrices.
		eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part.
		A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part.
	docs.sun.com /source/816-2461/stgevc.html (1042 words)

	Abstract of Defensio Talk of Tuerker Biyikoglu (Site not responding. Last check: 2007-11-04)
		We consider the adjacency, Laplacian, and generalized Laplacian (a symmetric matrix with non-positive off-diagonal elements) of a graph.
		We characterize for a tree: the maximal number of the strong nodal domains of an eigenvector corresponding to the k -th eigenvalue.
		Finally we prove the conjecture that the rank of the adjacency matrix of a cograph is equal to the number of distinct nonzero columns of the adjacency matrix.
	www.tbi.univie.ac.at /papers/Defensio/tuerker_abst.html (347 words)

	lapack-d/dtgevc.html (Site not responding. Last check: 2007-11-04)
		The j-th generalized left and right eigenvectors are y and x, resp., such that: H H y (A - wB) = 0 or (A - wB) y = 0 and (A - wB)x = 0 H Note: the left eigenvector is sometimes defined as the row vector y but DTGEVC computes the column vector y.
		It must be block upper triangular, with 1-by-1 or 2-by-2 blocks on the diagonal, the 1-by-1 blocks corresponding to real generalized eigenvalues and the 2-by-2 blocks corresponding to complex generalized eigenvalues.
		When finding left eigenvectors, the matrix in question is the transpose of the one in storage, so the rowwise method then actually accesses columns of A and B at each step, and so is the preferred method.
	www.math.utah.edu:8080 /software/lapack/lapack-d/dtgevc.html (1022 words)

	Eigenvector/Null-Space Purification
		It is clear that the goal is to prevent components in from corrupting the vectors Thus to begin, the starting vector should be of the form If a final approximate eigenvector has components in they may be purged by replacing and then normalizing.
		One is to note that if with then and the approximate eigenvector is replaced with the improved approximation where.
		Thus, in order to completely eradicate components from one must multiply by where is equal to the dimension of the largest Jordan block corresponding to a zero eigenvalue of.
	www.caam.rice.edu /software/ARPACK/UG/node54.html (433 words)

	Pellionisz: Tensor Model of Gaze Control
		Since, for Eigenvector input, the output differs only by the Eigenvalue coefficient, it follows that the Eigenvectors of the covariant and contravariant metric are the same, with reciprocal Eigenvalues.
		The extraocular muscle system is a motor mechanism that, by means of generating an ideally equal but opposite eye movement, compensates for head movements and thus maintains the direction of the gaze in a moving head.
		In this sense, both the contravariant generation of the invariant and the covariant vectorial measurement of the same is available in the vestibulo-collicular sensorimotor reflex (cf., the scheme in Pellionisz (1984b) and in Chapter 15).
	www.usa-siliconvalley.com /inst/pellionisz/berthoz/berthoz.html (8696 words)

	[No title]
		In the examples I've considered, there is always a single pair; in the general case, one should take all eigenvectors corresponding to 0 singular values (this just represents multiplicity), and check that there were as many of them as there were multiples of the eigenvalue.
		The point is to provide a means for determining which generalized eigenvectors belong with which eigenvectors, and put them all together, creating the A = S J S^(-1) matrix decomposition.
		In the event that the eigenvectors are distinct, then J is just a diagonal matrix of eigenvalues; otherwise, it contains 1s in the super-diagonal of the matrix.
	www.nku.edu /~longa/xlispstat/eigs/eigs.lsp (1135 words)

	dtgevc(3) (Site not responding. Last check: 2007-11-04)
		The right generalized eigenvector x and the left generalized eigenvector y of (A,B) corresponding to a generalized eigenvalue w are defined by: (A - wB) * x = 0 and y*H (A - wB) = 0 where y**H denotes the conjugate tranpose of y.
		If (A,B) was obtained from the generalized real-Schur factorization of an original pair of matrices (A0,B0) = (QAZ*H,QBZH), then ZX and QY are the matrices of right or left eigenvectors* of A. A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal blocks.
		Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part.
	h18000.www1.hp.com /math/documentation/cxml/dtgevc.3lapack.html (1097 words)

	Finding the Jordan Canonical Form of a Matrix (Site not responding. Last check: 2007-11-04)
		This matrix S will be made up of eigenvectors and "generalized eigenvectors" as we shall see.
		v(1)=the first eigenvector, v(2)=generalized eigenvector to go with v(1), and v(3) is the second eigenvector.
		This does not necessarily mean that B does not have a full complement of eigenvectors.
	www.ma.iup.edu /projects/CalcDEMma/JCF/jcf5.html (370 words)

	No Title
		In the general case, the diagonal blocks are ``nearly diagonal,'' and J is block diagonal.
		These statements cover the situation in which there may be more than one generalized eigenvector associated with a particular Jordan block.
		A general rule emerges from these observations: The algebraic multiplicity of an eigenvalue is the number of times that eigenvalue appears on the diagonal, while the geometric multiplicity of an eigenvalue is the number of different blocks that contain the eigenvalue.
	www-math.cudenver.edu /~wbriggs/5718s01/notes_jordan/notes_jordan.html (1438 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		A generalized eigenvalue of the pair (A,B) is, roughly speaking, a scalar of the form lambda=alpha/beta such that the matrix A-lambda*B is singular.
		The left generalized eigenvectors u(j) are stored in the columns of VL in the order of their eigenvalues.
		The right generalized eigenvectors v(j) are stored in the columns of VR in the order of their eigenvalues.
	www.netlib.org /lapack95/DOC/la_ggev.txt (628 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		Does "A" have a complete set of > generalized eigenvectors that are solutions of > Ax = \lambda Bx (If you're talking about hermitian operators in an infinite-dimensional Hilbert space, I'll interpret "complete set of eigenvectors" as physicists' terminology for the Spectral Theorem.) If B is positive definite, yes.
		For then B has a positive definite square root, and the generalized eigenvector problem is equivalent to the ordinary eigenvector problem B^(-1/2) A B^(-1/2) y = lambda y where B^(-1/2) A B^(-1/2) is hermitian.
		Note that the generalized eigenvectors x are related to the eigenvectors y of this operator by x = B^(-1/2) y, so these are not orthogonal in the ordinary inner product.
	www.math.niu.edu /~rusin/known-math/99/gen_eig (274 words)

	[No title]
		A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singular.
		The right generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j).
		The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies u(j)*H A = lambda(j) * u(j)*H B where u(j)**H is the conjugate-transpose of u(j).
	www.ibiblio.org /gferg/ldp/man/manl/zggev.l.html (636 words)

	cggevx(l): compute for pair of N-by-N complex ... - Linux man page
		The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies u(j)*H A = lambda(j) * u(j)*H B. where u(j)**H is the conjugate-transpose of u(j).
		If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues.
		If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues.
	www.die.net /doc/linux/man/manl/cggevx.l.html (946 words)

	[No title]
		* * The right generalized eigenvector x and the left generalized * eigenvector y of (A,B) corresponding to a generalized eigenvalue * w are defined by: * * (A - wB) * x = 0 and y*H (A - wB) = 0 * * where y**H denotes the conjugate tranpose of y.
		If (A,B) was obtained from the generalized real-Schur * factorization of an original pair of matrices * (A0,B0) = (QAZ*H,QBZH), then ZX and QY are the matrices of right or left eigenvectors of * A. * A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal * blocks.
		Corresponding to each 2-by-2 diagonal block is a complex * conjugate pair of eigenvalues and eigenvectors; only one * eigenvector of the pair is computed, namely the one corresponding * to the eigenvalue with positive imaginary part.
	www.cs.utk.edu /~eanderso/bad/stgevc.f (1787 words)

	Advanced Numerical Methods: Features
		Inverse-free methods based on the generalized eigenvector and the generalized Schur decompositions
		System identification from the impulse responses, frequency responses, and generic input-output data
		Computation of the generalized eigenvalues, generalized eigenvectors, and generalized Schur decomposition
	www.wolfram.com /products/applications/anm/features.html (209 words)

	lapack-s/stgevc.html (Site not responding. Last check: 2007-11-04)
		The j-th generalized left and right eigenvectors are y and x, resp., such that: H H y (A - wB) = 0 or (A - wB) y = 0 and (A - wB)x = 0 H Note: the left eigenvector is sometimes defined as the row vector y but STGEVC computes the column vector y.
		B (input) REAL array, dimension (LDB,N) The other of the pair of matrices whose generalized eigenvectors are to be computed.
		VL (input/output) REAL array, dimension (LDVL,MM) On exit, the left eigenvectors (column vectors -- see the note in "Purpose".) Real eigenvectors take one column, complex take two columns, the first for the real part and the second for the imaginary part.
	www.math.utah.edu:8080 /software/lapack/lapack-s/stgevc.html (1017 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		The right generalized eigenvector x and the left generalized eigenvector y of (A,B) corresponding to a generalized eigenvalue w are defined by:.br (A - wB) * x = 0 and y*H (A - wB) = 0.br where y**H denotes the conjugate tranpose of y.
		If (A,B) was obtained from the generalized real-Schur factorization of an original pair of matrices.br (A0,B0) = (QAZ*H,QBZH),.br then ZX and QY are the matrices of right or left eigenvectors* of A.
		Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one.br eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part.
	www.helsinki.fi /atk/unix/sun-manuals/WS6U1/man/man3p/dtgevc.3p (997 words)

	Sun Performance Library Reference: 1 - LAPACK Subroutines
		On entry, the first of the pair of matrices whose generalized eigenvalues (and Schur vectors) are to be computed.
		On exit, the generalized Schur form of A. Note: to avoid overflow, the Frobenius norm of A should be less than the overflow threshold.
		On exit, the generalized Schur form of B. Note: to avoid overflow, the Frobenius norm of B should be less than the overflow threshold.
	www.math.ias.edu /mcomputer/SUNWspro/html_docs/perflib/perflib_ref/chapter1.doc17.html (958 words)

	ETNA Volume 7, 1998
		Common features of these eigenvalue problems are (1) the number of eigenvalues required is small relative to the size of the matrices and (2) the matrix systems are often very sparse or structured.
		A standard technique is to formulate a spectral transformation such as shift-invert in order to enhance the convergence to the eigenvalues and eigenvectors of interest.
		On May 14-16, 1997, Paul Plassmann and Richard B. Lehoucq hosted a workshop at Argonne National Laboratory called "The Use of Iterative Methods for Large Scale Eigenvalue Problems." The workshop generated interest within the numerical linear algebra community and was attended by a number of its leading researchers.
	etna.mcs.kent.edu /vol.7.1998/index.html (397 words)

	[No title]
		If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvec tors of (A,B), or the products ZX and/or QY, where Z and Q are input orthogonal matrices.
		If (A,B) was obtained from the generalized real-Schur factorization of an origi nal pair of matrices (A0,B0) = (QAZ*H,QBZH), then ZX and Q*Y are the matrices of right or left eigen vectors of A. A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal blocks.
		Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corre sponding to the eigenvalue with positive imaginary part.
	www.ibiblio.org /gferg/ldp/man/manl/stgevc.l.html (957 words)

	ICML 2005 in Bonn, Germany: International Conference on Machine Learning
		The central idea underlying these methods is that although natural data is typically represented in very high dimensional spaces, the process generating the data is often thought to have relatively few degrees of freedom.
		A natural mathematical characterization of this intuition is to model the data as lying on or near a low dimensional manifold embedded in a higher dimensional space.
		Solutions of K-means clustering are given by the eigenvector of the inner-product kernel matrix whereas solutions of the spectral clustering objective are given by the generalized eigenvector of the same matrix.
	icml.ais.fraunhofer.de /tutorials.php (1926 words)

	Spectral Transformations R. Lehoucq and D. Sorensen
		Quite often, however, the wanted eigenvalues may not be well separated or located in the interior of the convex hull of eigenvalues.
		In these situations, iterative projection methods need many steps to generate acceptable approximations, to these eigenvalues, if they converge at all.
		The SI is extremely effective in terms of iteration steps (that is the dimension of the subspace) and should be used whenever possible.
	www.cs.utk.edu /~dongarra/etemplates/node84.html (420 words)

	zgegv (Site not responding. Last check: 2007-11-04)
		A left generalized eigenvector is a vector l such that (A - w B)**H l = 0.
		A (input/workspace) COMPLEX16 array, dimension (LDA, N) On entry, the first of the pair of matrices whose generalized* eigenvalues and (optionally) generalized eigenvectors are to be computed.
		B (input/workspace) COMPLEX16 array, dimension (LDB, N) On entry, the second of the pair of matrices whose generalized* eigenvalues and (optionally) generalized eigenvectors are to be computed.
	www.nacse.org /demos/lapack/routines/zgegv.html (629 words)

	Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations
		n, one must find the complete set of eigenvalues and eigenvectors of the matrix A. If A has repeated eigenvalues, one might also need to compute their generalized eigenvectors.
		The Eigenvectors and Generalized Eigenvectors of A Form a Basis of R
		General Solution: The Eigenvectors and Generalized Eigenvectors of A Form a Basis of R
	www.egwald.com /linearalgebra/lineardifferentialequations.php (1571 words)

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