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Topic: Invariant subspace

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	Invariant Subspaces (Site not responding. Last check: 2007-09-17)
		On page 76 is that if we can decompose the space V into a direct sum of invariant subspaces under T, we can pose the simpler problem of studying what T does to the invariant subspaces.
		The subspace 0, V, null T and range T are all invariant subspaces of a linear operator T on V.
		Invariant subspaces of dimension 1 are spanned by a single vector, which is an eigenvector of some eigenvalue.
	www.rpi.edu /~piperb/linalg/lecture/10-19/node2.html (197 words)

	2 An introduction to nonlinear optimization problem structure
		However, sparsity is not the most important phenomenon associated with a nonlinear function; that role is played by invariant subspaces.
		The importance of invariant subspaces is that nonlinear information is not required for a function in this subspace.
		We are particularly interested in functions which have large (as a percentage of the overall number of variables) invariant subspaces.
	www.numerical.rl.ac.uk /lancelot/sif/node2.html (417 words)

	Invariant subspace - Wikipedia, the free encyclopedia
		The invariant subspace problem concerns the case where V is a separable Hilbert space over the complex numbers, of dimension > 1, and T is a bounded operator.
		More generally, invariant subspaces are defined for sets of operators (operator algebras, group representations) as subspaces invariant for each operator in the set.
		If a subspace W of V is invariant with respect to all these transformations, then it is a subrepresentation and the group G acts on W in a natural way.
	en.wikipedia.org /wiki/Invariant_subspace (465 words)

	PlanetMath: invariant
		is a well-defined transformation of the invariant subset:
		In this case we say that a subset is invariant, if it is invariant with respect to all elements of the family.
		This is version 5 of invariant, born on 2002-02-22, modified 2002-02-22.
	planetmath.org /encyclopedia/Invariant.html (81 words)

	[No title] (Site not responding. Last check: 2007-09-17)
		We consider the behaviour of attractors near invariant subspaces on varying a parameter that does not preserve the dynamics in the invariant subspace but is otherwise generic, in a smooth dynamical system.
		If there is chaos in the invariant subspace that is not structurally stable, this has the effect of ``blurring out'' blowout bifurcations over a range of parameter values that we show can have positive measure in parameter space.
		Associated with such blowout bifurcations are bifurcations to attractors displaying a new type of intermittency that is phenomenologically similar to on-off intermittency, but where the intersection of the attractor by the invariant subspace is larger than a minimal attractor.
	www.maths.qmw.ac.uk /~eoc/papers/abstracts/nonnormal.html (208 words)

	PlanetMath: invariant subspace
		is an invariant subspace, then the restriction of
		Cross-references: transformation, matrix, vectors, basis, restriction, subspace, vector space, linear transformation
		This is version 5 of invariant subspace, born on 2002-02-15, modified 2005-08-02.
	planetmath.org /encyclopedia/InvariantSubspace.html (65 words)

	Direct Sum of Vector Spaces (Site not responding. Last check: 2007-09-17)
		A subspace is invariant under a transformation if that subspace is mapped into itself.
		Show that the span and intersection of two invariant subspaces is an invariant subspace.
		Subspaces are complementary if they are invariant and their direct sum produces the original space.
	www.mathreference.com /la-jf,sum.html (493 words)

	subspace iteration in the similarity reduction
		91], where the size of the vector subspace is increased by one and a change of coordinate system is made at each step of the algorithm.
		We will see that the reduction algorithm from a symmetric to a semiseparable matrix can be interpreted as such a kind of subspace iteration, where the dimension of the subspace grows by one at each step of the algorithm.
		This explains why in the numerical examples (see Chapter 6), the lower right block already gives a good estimate of the largest eigenvalues, since they are connected to a subspace on which the subspace iteration is performed most.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node58.html (1256 words)

	Math 5718 Notes 3
		Recall that the author of the text approaches the eigenvalue problem through invariant subspaces, without using the traditional definition of an eigenvalue (involving determinants and characteristic polynomials).
		In this case, the double eigenvalue has an invariant subspace of dimension 1.
		In this case, there is an invariant subspace of dimension 2, with a basis
	www-math.cudenver.edu /~wbriggs/5718s01/notes3/notes3.html (512 words)

	Structure of the Eigenvalue Problem
		A subspace of is called an invariant subspace of
		Invariant subspaces are central to the methods developed here.
		In the symmetric (Hermitian) case, invariant subspaces corresponding to distinct eigenvalues are orthogonal to each other and completely decouple the action of the matrix (as an operator on.
	www.caam.rice.edu /software/ARPACK/UG/node46.html (1176 words)

	Invariant theory and generalized Casimir operators
		Different sets of generalized commuting Casimir operators generate different orthonormal bases in the invariant subspace; the overlaps between the eigenvectors of different commuting sets of generalized Casimir operators are called invariant coefficients.
		We show that Racah coefficients are special cases of invariant coefficients in which the generalized Casimir operators have been chosen with respect to a definite coupling scheme in the tensor product.
		Eigenvectors in the six-dimensional invariant subspace are computed for different sets of generalized Casimir operators and invariant coefficients, including Racah coefficients.
	stacks.iop.org /0305-4470/34/8237 (532 words)

	Invariant subspaces
		I'm not sure whether it's necessary for an operator to have eigenvalues in order for it to have an invariant subspace, but it is sufficient (although matt grime's post suggests that it is necessary).
		I suspect his teacher referred to the famous open problem: the "invariant subspace problem", posed over infinite dimensional complex Hilbert spaces where determinants are unavailable.
		I.e, there does exist a bounded linear operator on a Banach space B with no invariant subspace except {0} and B. (A Banach space is a linear space with a concept of length, and in which Cauchy's convergence criterioon is satisifed.
	www.physicsforums.com /showthread.php?p=384371 (1062 words)

	METU MATHEMATICS DEPARTMENT
		Abstract: The investigation of invariant subspaces is a natural first step in the attempt to understand the structure of operators.
		The powerful structure theorems that are known for finite-dimensional operators (the Jordan form) and normal operators (the spectral theorem) provide, in essence, decompositions into invariant subspaces of special kinds.
		Some recent advances will be presented and a new result on the existence of an invariant ideal for a family of positive operators on locally convex solid Riesz spaces will be given.
	www.math.metu.edu.tr /seminars/caglar_abstract.shtml (163 words)

	Schur's Decomposition
		a one-dimensional subspace that is invariant with respect to the multiplication by the matrix A on the left.
		of the column-vectors of the matrix X is invariant with respect to the multiplication by the matrix A on the left.
		of the row-vectors is invariant with respect to the multiplication by the matrix B on the right.
	www.cs.ut.ee /~toomas_l/linalg/lin1/node17.html (645 words)

	Invariant Subspace Computation: A Geometric Approach (Site not responding. Last check: 2007-09-17)
		Its iterates are pairs of p-dimensional subspaces of R^n and it converges locally cubically to the pairs of left-right nondefective spectral eigenspaces of arbitrary square matrices.
		Using the representation of p-dimensional subspaces of R^n by n-by-p matrices that span the subspace, we derive formulas for essential differential-geometric objects on the Grassmann manifold, including the O_n-invariant metric and the associated Riemannian connection and geodesics.
		We propose a simple way to select the deformation parameter in the course of the iteration so as to benefit from the global convergence properties of the gradient descent flow while preserving the cubic convergence rate of the pure Newton method.
	www.montefiore.ulg.ac.be /systems/Publi/absil_thesis.htm (470 words)

	lapack-z/ztrsen.html (Site not responding. Last check: 2007-09-17)
		Optionally the routine computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.
		First we compute R so that P = (I R) n1 (0 0) n2 n1 n2 is the projector on the invariant subspace associated with T11.
		When SEP is small, small changes in T can cause large changes in the invariant subspace.
	www.math.utah.edu:8080 /software/lapack/lapack-z/ztrsen.html (608 words)

	Invariant Subspaces and Condition Numbers
		Then the first j columns of Z span the right invariant subspace of A corresponding to
		xTRSEN moves a selected subset of the eigenvalues of a matrix T in Schur form to the upper left corner of T, and optionally computes the condition numbers of their average value and of their right invariant subspace.
		These are the same as the condition numbers of the average eigenvalue and right invariant subspace of the original matrix A from which T is derived.
	www.netlib.org /lapack/lug/node52.html (291 words)

	Introduction
		The symmetric invariant subspace decomposition algorithm (SYISDA) for an
		Note that the PRISM algorithm is able to utilize matrix multiplication by using the mathematical fact that applying a polynomial to matrix changes its eigenvalues but not its invariant subspaces, i.e., eigenvectors.
		However, if one is only interested in finding an orthogonal basis for the subspace spanned by a certain set of eigenvectors, there is no need to expend the effort to compute all eigenvalues, and one can terminate when the subspace corresponding to the desired eigenvalue range has been identified.
	www-unix.mcs.anl.gov /prism/lib/intro.html (414 words)

	ZSP: Spin Up and Split
		If a proper invariant subspace is found, the program can either write out this space or `split' the representation, i.e., calculate the action of the matrices on both the subspace and its quotient.
		This is done by retaining always a semi-echelon form of the space found so far, and multiplying the rows of this by all the generators, appending the (Gaussian-eliminated) result to the space if it is not already in, and continuing until all vectors in the semi-echelon form have been treated.
		The maximal number of seed vectors generated and used for splitting is the number of one dimensional subspaces of the seed space.
	www.math.rwth-aachen.de /homes/Meataxe/htmldoc/node43.html (1338 words)

	Invariant Manifolds
		The eigenspaces of a linear flow or map (i.e., the spaces formed by the eigenvectors of A) are invariant subspaces of the dynamical system.
		Moreover, the dynamics on each subspace are determined by the eigenvalues of that subspace.
		and eigenvectors (1,-1,0), (0,0,1), (2,1,0), and the flow on the invariant manifolds is illustrated in Figure 4.10.
	www.drchaos.net /drchaos/Book/node118.html (200 words)

	ZTRSEN
		Specifies whether condition numbers are required for the cluster of eigenvalues (S) or the invariant subspace (SEP):
		is the projector on the invariant subspace associated with T11.
		The reciprocal condition number of the right invariant subspace spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. SEP is defined as the separation of T11 and T22:
	www.math.ucla.edu /computing/docindex/lapack-manpages-man-1259.html (710 words)

	Specifying an Eigenproblem
		This is typically done by computing an approximate right (and perhaps left) invariant subspace corresponding to eigenvalues in the desired region.
		The user can also compute the right (and perhaps left) eigenvectors in the computed invariant subspace.
		For the eigenvalues that are clustered together, the user may choose to compute the associated invariant subspace, since in this case the individual eigenvectors can be very ill-conditioned, while the invariant subspace may be less so.
	www.cs.utk.edu /~dongarra/etemplates/node54.html (225 words)

	Cubically convergent iterations for invariant subspace computation (Site not responding. Last check: 2007-09-17)
		We propose a Newton-like iteration that evolves on the set of fixed dimensional subspaces of $\rr^n$ and converges locally cubically to the invariant subspaces of a symmetric matrix.
		This iteration is compared in terms of numerical cost and global behaviour with three other methods that display the same property of cubic convergence.
		Shorter version focusing on Newton methods: ``A Newton algorithm for invariant subspace computation with large basins of attraction'', P.-A. Absil, R. Sepulchre, P. Van Dooren and R. Mahony, Proceedings of the 42nd IEEE Conference on Decision and Control, December 9-12, 2003, Hyatt Regency Maui, Hawaii, USA, pp.
	www.montefiore.ulg.ac.be /systems/Publi/Grass_iter.htm (148 words)

	[No title]
		CTRSEN reorders the Schur factorization of a complex matrix A = QTQH, so that a selected cluster of eigen values appears in the leading positions on the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace**.
		JOB (input) CHARACTER1 Specifies whether condition numbers are required for the cluster of eigenvalues (S) or the invari* ant subspace (SEP): = 'N': none; = 'E': for eigenvalues only (S); = 'V': for invariant subspace only (SEP); = 'B': for both eigenvalues and invariant subspace (S and SEP).
		SEP (output) REAL If JOB = 'V' or 'B', SEP is the estimated recipro cal condition number of the specified invariant subspace.
	www.ibiblio.org /gferg/ldp/man/manl/ctrsen.l.html (804 words)

	About "PRISM (Parallel Research on Invariant Subspace Methods)" (Site not responding. Last check: 2007-09-17)
		A project the goal of which is to develop infrastructure and algorithms for the parallel solution of eigenvalue problems.
		PRISM is currently investigating a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA).
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/view/10096.html (88 words)

	Atlas: The Existence of Translation Invariant Subspaces of Symmetric Self-Adjoint Sequence Spaces on $\bf Z$ by Aharon ... (Site not responding. Last check: 2007-09-17)
		If X is a translation invariant Banach space of complex sequences on the integer group Z, then from the point of view of harmonic analysis, it is natural to study the (closed) translation invariant subspaces of X. The first problem that arises in this context, is whether X has any nontrivial translation invariant subspace.
		The proof of the theorem is based on the following intermediate result, which was inspired by a paper of Simonic [2].
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-20.
	atlas-conferences.com /cgi-bin/abstract/caeo-20 (382 words)

	On the discrete Conley index in the invariant subspace (ResearchIndex) (Site not responding. Last check: 2007-09-17)
		On the discrete Conley index in the invariant subspace (ResearchIndex)
		On the discrete Conley index in the invariant subspace (1997)
		Abstract: We present theorems concerning the relations between the discrete homotopy Conley index in the affine invariant subspace and the index calculated in the entire space or in the half space.
	citeseer.ist.psu.edu /66796.html (357 words)

	Citebase - A Reducing of the Invariant Semidefinite Subspace Problem for Krein Noncontraction to such a Problem for ...
		A Reducing of the Invariant Semidefinite Subspace Problem for Krein Noncontraction to such a Problem for Krein Isometry
		Let J be a period-2 unitary operator (some people say J is reflection operator or reflection symmetry) and U be a linear operator.
		If every J-isometry has nontrivial positive invariant subspace then every J-noncontraction has such a subspace.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9909101 (200 words)

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