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Topic: Eigenvalue, eigenvector and eigenspace

	Eigenvalue
		An eigenspace of a given transformation is the span of the eigenvectors of that transformation with the same eigenvalue, together with the zero vector (which has no direction).
		Eigenvalues are often introduced in the context of linear algebra or matrix theory.
		The admittable eigenvalues are governed by the length of the string and determine the frequency of oscillation.
	www.algebra.com /algebra/college/linear/Eigenvalue.wikipedia (3866 words)

	eigenspace@Everything2.com
		An eigenspace of an n x n matrix is the subspace spanned by the eigenvectors corresponding to a given eigenvalue.
		Essentially, an eigenspace is the null space of a matrix A-λI, where λ is an eigenvalue of the matrix A and I is the identity matrix.
		Since all eigenvectors for a given eigenvalue are linearly independent, the dimension of the eigenspace is equal to the number of linearly independent vectors that satisfy (A-λI) v = 0 which follows from the definition of an eigenvector.
	www.everything2.org /node/1938686 (479 words)

	Eigenvector info here at en.12-year.info (Site not responding. Last check: )
		parallel to that vector - are with eigenvectors, with the synonymous eigenvalue.
		This is the characteristic polynomial of A: the eigenvalues of a matrix are the zeros of its characteristic polynomial.
		The supereminent eigenvector of a graph is used to breadth the centrality of its vertices.
	en.12-year.info /Eigenvector (5249 words)

	EIGENVALUE, EIGENVECTOR AND EIGENSPACE FACTS AND INFORMATION (Site not responding. Last check: )
		The geometric multiplicity of an eigenvalue is the dimension of the associated eigenspace.
		This is the characteristic polynomial of ''A'': the eigenvalues of a matrix are the zeros of its characteristic polynomial.
		The principal eigenvector of a graph is used to measure the centrality of its vertices.
	www.feefriend.com /Eigenvalue,_eigenvector_and_eigenspace (3821 words)

	No Title
		Eigenvalues and eigenvectors are needed for the analysis of many types of mathematical models.
		The problem of finding the eigenvalues and eigenvectors for a square real matrix A is a very different kind of problem compared to the problems that have been solved so far in this course, where the primary tool has been elementary row operations.
		In this case, the eigenvalues of A (and D) are the diagonal entries of D.
	www.math.wsu.edu /faculty/genz/220v/lessons/l10/l10.html (350 words)

	Eigenvector : Eigenspace (Site not responding. Last check: )
		In linear algebra, the eigenvectors (from the German eigen meaning "own") of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves.
		In classical mechanics the eigenvectors of the governing equations typically correspond to natural modes of vibration in a body, and the eigenvalues to their frequencies.
		This is the characteristic polynomial of A; its zeros are precisely the eigenvalues of A.
	www.termsdefined.net /ei/eigenspace.html (969 words)

	Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia
		Nowadays, eigenvalues are usually introduced in the context of matrix theory.
		An eigenspace is a space consisting of all eigenvectors which have the same eigenvalue, along with the zero (null) vector, which itself is not an eigenvector.
		The principal eigenvector of a transformation is the eigenvector with the largest corresponding eigenvalue.
	en.wikipedia.org /wiki/Eigenvalue,_eigenvector_and_eigenspace (5176 words)

	Eigenvalue-Eigenstate-Eigenvector
		The eigenvector of a given linear transformation is that vector which is not changed by that transformation.
		In the same way, an eigenvalue is a magnitude proportion of the vector that does not change in the transformation, and the eigenspace of a transformation is the set of all eigenvectors (plus the zero vector) that have the same eigenvalue.
		Thus an eigenspace is a subspace of a total vector space.
	www.iscid.org /encyclopedia/Eigenvalue-Eigenstate-Eigenvector (208 words)

	Eigenvalue, eigenvector and eigenspace at Decipedia.com — The simple online encyclopedia!
		Eigenvectors of transformations are vectors which are either left unaffected or simply multiplied by a scale factor after the transformation.
		An eigenspace is a Mathematics_and_space_">space consisting of all eigenvectors which have the same eigenvalue, along with the zero(null) vector which itself is not an eigenvector.
		The algebraic multiplicity of an eigenvalue λ of A is the order of λ as a zero of the characteristic polynomial of A; in other words, if λ is one root of the polynomial, it is the number of factors (t − λ) in the characteristic polynomial after factorization.
	www.decipedia.com /article/Eigenvalue,_eigenvector_and_eigenspace (4334 words)

	No Title
		Eigenvalues and eigenvectors are needed for the analysis of many types of mathematical models.
		The problem of finding the eigenvalues and eigenvectors for a square real matrix A is a very different kind of problem compared to the problems that have been solved so far in this course, where the primary tool has been elementary row operations.
		In this case, the eigenvalues of A (and D) are the diagonal entries of D.
	www.sci.wsu.edu /math/faculty/genz/220v/lessons/l10/l10.html (350 words)

	Some eigennotes
		An eigenvalue from the first block will have a row eigenvector which has only a few non-zeroes (at most as many as the size of the block), and a large number of training zeroes.
		Almost all of the eigenvectors in the space will have fewer zeroes than either of the two that we got from the blocks, and so there is something special about the original pair.
		The main purpose is that we can identify particular properties of the eigenvectors based either on their symmetries or on their leading/trailing zeroes, and thus acquire insight into the structure of the system that we are analysing.
	www.cl.cam.ac.uk /~mas33/moreeigens.htm (630 words)

	finrevII.html (Site not responding. Last check: )
		The eigenvalues of A are the roots of the characteristic polynomial
		Eigenvectors of A belonging to distinct eigenvalues of A are independent.
		Suppose that any k eigenvectors belonging to distinct eigenvalues of A are independent and take any set of k+1 eigenvectors belonging to distinct eigenvalues.
	www.msc.uky.edu /carl/ma322/spr2001/html/finrevII1.html (513 words)

	PlanetMath: eigenvalue
		A linear operator can have several eigenvalues (but no more than the dimension of the space).
		Cross-references: linearly independent, dimension, eigenvalues, kernel, vector, scalar, linear operator, vector space
		This is version 5 of eigenvalue, born on 2003-10-15, modified 2004-06-03.
	planetmath.org /encyclopedia/Eigenvector2.html (78 words)

	The eigenvalue problem
		But we may also find that only one eigenvector is associated with a multiple root and that the operator does not have a basis of eigenvectors.
		The eigenvalues of a unitary operator are complex numbers of unit magnitude.
		denotes the degree of degeneracy of the eigenvalue a
	electron6.phys.utk.edu /qm1/modules/m3/eigenvalue.htm (951 words)

	eigenvalues - Search Results - MSN Encarta
		Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots...
		Eigenvalues Eigenvectors calculation in your VB.NET, C#, C++ source code...
		Eigenvalues and Eigenvectors calculation in just one line of your source code
	encarta.msn.com /eigenvalues.html (132 words)

	[No title]
		EIGENVECTORS In mathematics, an EIGENVECTOR of a transformation is a vector whose direction is unchanged by that transformation.
		The EIGENSPACE for a factor is the set of eigenvectors with that factor as eigenvalue.
		Solving for the eigenvalue I found the quadratic equation had a negative number under the radical (square root sign), so there were no real roots.
	www.well.com /~abs/Cyb/4.669211660910299067185320382047/c3m_0601.txt (13265 words)

	MathDL \| Tool Building: Web-based Linear Algebra Modules
		We consider an associated cognitively-guided activity as the means of forcing students to come to grips with both the computational process and the underlying geometric relationship between eigenvalues and eigenvectors.
		This sample activity provides a guided exploration of eigenvalues and eigenvectors of a particular matrix and includes a set of questions that can be asked for any other matrix, as well as some general questions about the tool and observations made from interacting with the tool.
		One of our goals in having students explore the geometric perspective associated with eigenvalues and eigenvectors is to get students to begin to recognize that computations cannot be taken at face value but need to be examined within context for validity.
	mathdl.maa.org /mathDL/4/?pa=content&sa=viewDocument&nodeId=560&pf=1 (8575 words)

	FAQTs - Knowledge Base - View Entry - Math: Matrix: Eigenvector: Eigenvalue: Operation: Calculate: How to calculate?
		If the result vector is such that only the length is changed (and not the direction, because the vector is rotated or translated), or thus in other words the result vector is parallel to the original vector, you have found an eigenvector.
		So its eigenvalues will be complex To quickly calculate this, from the trace and determinant rule (see video lecture Gilbert Strang) lambda1 + lambda2 = 0 + 0 = 0 lambda1.
		--- This is because the substitution of an eigenvalue lambda in the eigenvector equation always gives a singular homogeneous linear system, and among its infinity of solutions you generally have to seek a 'simple' solution (with small integer values (if this is possible)).
	www.faqts.com /knowledge_base/view.phtml/aid/39011/fid/1856 (1385 words)

	Review of Chapters 5 and 6 and loose ends
		The geometric multiplicity is the dimension of the eigenspace for the eigenvalue.
		The columns of P are the eigenvectors from a linearly independent set of eigenvectors for A.
		The matrix D is a diagonal matrix, and the entries on the diagonal are the corresponding eigenvalues of A.
	www.math.uiuc.edu /~johnmaki/ch6review.html (1579 words)

	11.1.6 Eigenvalue Recursive Bisection
		The highest eigenvalue of N must be chosen from the space of the lowest eigenvalue of E.
		This eigenstate is the second eigenvector of the Laplacian matrix of the graph.
		A closely related eigenvector method [Barnes:82a], [Boppana:87a] is based on the second highest eigenvector of the adjacency matrix of the graph, rather than the second lowest eigenvector of the Laplacian matrix.
	www.netlib.org /utk/lsi/pcwLSI/text/node254.html (1115 words)

	PlanetMath: linear algebra
		Important examples of such techniques include: Gaussian elimination, the method of least squares, LU factorization, QR decomposition, Gram-Schmidt orthogonalization, singular value decomposition, and a number of iterative algorithms for the calculation of eigenvalues and eigenvectors.
		Obstructions: imaginary eigenvalues, nilpotent transformations, classification of 2-dimensional real transformations.
		Eigenvalue problems: singular value decomposition, Gauss and Jacobi-Siedel iterative algorithms.
	planetmath.org /encyclopedia/linearalgebra.html (481 words)

	eigenvector - Wiktionary
		(linear algebra) A vector that is not rotated under a given linear transformation; a left or right eigenvector depending on context.
		(physics, engineering) A right eigenvector; a nonzero vector x such that, for a particular matrix A, Ax = λx for some scalar λ which is its eigenvalue and an eigenvalue of the matrix.
		This page was last modified on 30 May 2008, at 15:00.
	en.wiktionary.org /wiki/eigenvector (73 words)

	Study Guide 9 (Site not responding. Last check: )
		Determine whether a given vector is an eigenvector of a square matrix, and find the corresponding eigenvalue.
		Determine whether a given scalar is an eigenvalue for a matrix, and find a basis for the associated eigenspace.
		Use the eigenvalues and eigenvectors of a matrix to diagonalize the matrix or determine that this isn't possible.
	www-math.cudenver.edu /~billups/courses/ma3191/study/study9.html (143 words)

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