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# Topic: Eigenvalue

 PlanetMath: eigenvalue problem Many problems in physics and elsewhere lead to differential eigenvalue problems, that is, problems where the vector space is some space of differentiable functions and where the linear operator involves multiplication by functions and taking derivatives. As a result, matrix eigenvalues are useful in statistics, for example in analyzing Markov chains and in the fundamental theorem of demography. This is version 18 of eigenvalue problem, born on 2002-01-14, modified 2006-06-15. planetmath.org /encyclopedia/EigenvalueProblem.html   (541 words)

 Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia In mathematics, and in particular in vector analysis, the eigenvalues, eigenvectors, and eigenspaces of a transformation (from a vector space to itself) are important properties of this transformation. The solution to the eigenvalue equation is N = exp(λt), the exponential function; thus that function is an eigenfunction of the differential operator d/dt with the eigenvalue λ. This is the characteristic polynomial of A: the eigenvalues of a matrix are the zeros of its characteristic polynomial. en.wikipedia.org /wiki/Eigenvalue   (4039 words)

 PlanetMath: eigenvalue Eigenvalues are of relatively little importance in connection with an infinite-dimensional vector space, unless that space is endowed with some additional structure, typically that of a Banach space or Hilbert space. Cross-references: Hilbert space, Banach space, structure, infinite-dimensional, conjugate, real, eigenvalue problem, variations, even, roots, determinants, matrix, basis, odd, algebraically closed, characteristic polynomial, variable, degree, polynomial, finite, map, operations, obvious, complex numbers, sequences, complex, dimension, infinite, surjective, injective, inverse, equivalent, linear algebra, finite dimensional, action, vector, scalar, linear mapping, Endomorphism, field, vector space This is version 11 of eigenvalue, born on 2002-01-19, modified 2006-06-09. planetmath.org /encyclopedia/Eigenvalue.html   (301 words)

 Eigenvalues and Eigenvectors - HMC Calculus Tutorial Eigenvalues and eigenvectors play a prominent role in the study of ordinary differential equations and in many applications in the physical sciences. Eigenvalues and eigenvectors of larger matrices are often found using other techniques, such as iterative methods. The eigenvalues of A are the roots of the characteristic polynomial www.math.hmc.edu /calculus/tutorials/eigenstuff   (368 words)

 Eigenvalue algorithm   (Site not responding. Last check: 2007-10-15) In linear algebra, one of the most important problems is designingefficient and stable algorithms for finding the eigenvalues of a matrix. The usual method for finding the eigenvalues of a small matrix is by using the characteristic polynomial. The characteristic polynomial, defined as det(A - λI), is a polynomial in λwhose roots are the eigenvalues of A. www.therfcc.org /eigenvalue-algorithm-137876.html   (321 words)

 Eigenvalues and Eigenvectors Remark 32 The eigenvalue of smallest magnitude of a matrix is the same as the inverse (reciprocal) of the dominant eigenvalue of the inverse of the matrix. Although musicians do not study eigenvalues in order to play their instruments better, the study of eigenvalues can explain why certain sounds are pleasant to the ear while others sound "flat" or "sharp." When two people sing in harmony, the frequency of one voice is a constant multiple of the other. Eigenvalue analysis is also used in the design of car stereo systems so that the sounds are directed correctly for the listening pleasure of the passengers and driver. ceee.rice.edu /Books/LA/eigen   (2757 words)

 Eigenvalue   (Site not responding. Last check: 2007-10-15) In matrix theory, an element in the underlying ring R of a square matrix A is called a right eigenvalue if there exists a nonzero column vector x such that Ax =λ x, or a left eigenvalue if there exists a nonzero row vector y such that yA''=''y λ. The algebraic multiplicity (or simply multiplicity) of an eigenvalue λ of A is the number of factors t -λ of the characteristic polynomial of A. The Second Eigenvalue of the Google Matrix This paper by Sepandar Kamvar and Taher Haveliwala proves analytically the second eigenvalue of the Google Matrix, which has implications for the PageRank algorithm. www.serebella.com /encyclopedia/article-Eigenvalue.html   (625 words)

 Eigen Value An eigenvalue of a n by n matrix A is a scalar c such that A*x = c*x holds for some nonzero vector x (where x is an n-tuple). The imaginary part of the EigenValue tells a "natural" frequency of cycles in the system. The eigenvalue is the a multiplicative factor for the transformation. c2.com /cgi/wiki?EigenValue   (666 words)

 V. - Thomas Pynchon Eigenvectors, eigenfunctions, and eigenvalues are just basic terms out of "matrix" theory (matrix in this sense being the rectangular or n x m arrays of values, a mathematical term--and matrices can be in n dimensions, lest the quibblers correct my n x m example!). An eigenvector is a fixed point of a linear map in a vector space - actually the vector's direction is fixed, it's length is multiplied by a scalar otherwise known as the `eigenvalue' associated with the eigenvector. So each eigenvalue is a root of the eigenfunction (equivalently, each eigenvalue is a zero of the annihilator functions). www.hyperarts.com /pynchon/v/extra/eigenvalue.html   (1325 words)

 Eigenvalues and Eigenvectors of a Matrix The trace of the matrix A, i.e., the sum of the elements on the main diagonal, is equal to the sum of all eigenvalues of the matrix A. The eigenvalues of both an upper triangular or a lower triangular matrix are the elements of the main diagonal, and only they. If all the eigenvalues of the matrices A and B are single and the matrices A and B are commutative, then they have common eigenvectors. www.cs.ut.ee /~toomas_l/linalg/lin1/node16.html   (955 words)

 Math 310 - Glossary of Linear Algebra Terms The algebraic multiplicity of an eigenvalue c of a matrix A is the number of times the factor (t-c) occurs in the characteristic polynomial of A. The eigenspace associated with the eigenvalue c of a matrix A is the null space of A - cI. The geometric multiplicity of an eigenvalue c of a matrix A is the dimension of the eigenspace of c. www.math.uic.edu /math310/glossary.html   (1060 words)

 MATH2071: LAB #12: The Eigenvalue Problem The Rayleigh quotient is a scalar value whose magnitude generally lies between the smallest and largest magnitudes of the eigenvalues of the matrix. For a bridge or support column, the eigenvalue might reveal the maximum load, and the eigenvector the shape of the object as it begins to fail under this load. Denote the smallest eigenvalue of A+3*I by mu. www.csit.fsu.edu /~burkardt/math2071/lab_12.html   (2014 words)

 Definition of Matrix eigenvalue problem In linear algebra, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. The characteristic polynomial, defined as $det\left(A - \lambda I\right), is a polynomial in\lambdawhose roots are the$eigenvalues of $A.$ However, finding the roots of the characteristic polynomial may be an ill-conditioned problem even when the underlying eigenvalue problem is well-conditioned. www.wordiq.com /definition/Matrix_eigenvalue_problem   (434 words)

 Eigenvalue Decomposition   (Site not responding. Last check: 2007-10-15) The word "eigen" is German and means "same"; this is appropriate because the vector x after the matrix multiplication is the same as the original vector x, except for the scaling factor. Once we have determined the eigenvalues of a particular matrix, we can start to discuss them in terms of their multiplicity. A particular eigenvalue's geometric multiplicity is defined as the dimension of the nullspace of λI−A. cnx.rice.edu /content/m2116/latest   (744 words)

 Eigenvalue - free-definition   (Site not responding. Last check: 2007-10-15) In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. In matrix theory, an element in the underlying ring R of a square matrix A is called a right eigenvalue if there exists a nonzero column vector x such that Ax=λx, or a left eigenvalue if there exists a nonzero row vector y such that yA=yλ. The algebraic multiplicity (or simply multiplicity) of an eigenvalue λ of A is the number of factors t-λ of the characteristic polynomial of A. www.free-definition.com /Eigenvalue.html   (388 words)

 Eigenvalues and Eigenfunctions   (Site not responding. Last check: 2007-10-15) Solutions exist for the time independent Schrodinger equation only for certain values of energy, and these values are called "eigenvalues*" of energy. The eigenvalue concept is not limited to energy. The eigenvalues qi may be discrete, and in such cases we can say that the physical variable is "quantized" and that the index i plays the role of a "quantum number" which characterizes that state. hyperphysics.phy-astr.gsu.edu /hbase/quantum/eigen.html   (175 words)

 Eigenvectors and eigenvalues   (Site not responding. Last check: 2007-10-15) An entirely different way that eigenvectors arise is in the theory of systems of differential equations, such as the equations which relate two populations, which we can think of as prey and predators. The eigenvalues are the roots of the polynomial, i.e., +1 and -1. There are two eigenvalues, but they are imaginary numbers, namely i and -i, the square roots of -1. www.mathphysics.com /calc/eigen.html   (3253 words)

 MATH2071: LAB #9: The Eigenvalue Problem   (Site not responding. Last check: 2007-10-15) For a bridge or support column, the largest eigenvalue might reveal the maximum load, and the eigenvector the shape of the object as it begins to fail under this load. (the Rayleigh quotient is the approximate eigenvalue) and the component of the vector Assume that the middle eigenvalue is near 2.5 and start with a vector of all 1's. www.math.pitt.edu /~sussmanm/2071Spring04/lab09/lab09.html   (3283 words)

 Nonsymmetric Eigenvalue Decomposition Classes The Nonsymmetric Eigenvalue Decomposition Classes are contained in Table 5. Encapsulates Hessenberg decomposition eigenvalue servers, which are used to construct eigenvalue decomposition objects. Encapsulates Schur decomposition eigenvalue servers, which are used to construct eigenvalue decomposition objects from Schur decompositions. www.roguewave.com /support/docs/sourcepro/lapackref/2-6.html   (155 words)

 Citations: Graph coloring using eigenvalue decomposition - Aspvall, Gilbert (ResearchIndex) Their application in graph drawing is discussed in [32, 37, 46, 51] The cost functions of a number of prominent combinatorial optimization problems, among them the TSP, graph bi partitioning, and certain spin glass models, are eigenfunctions of graphs associated with search heuristics for these.... In the case of degenerate eigenvalues the situation becomes even more di#cult because the number of nodal domains may vary considerably depending on which vector from the m k dimensional eigenspace of # k is chosen. For a precise description of the physical problem the reader is referred to the paper by Knallmann and Quellmalz [5] Their paper also proposes a simulated annealing algorithm for the approximate solution of the frequency assignment problem. citeseer.ist.psu.edu /context/379778/0   (2367 words)

 Symmetric Eigenvalue Decomposition Classes Encapsulates the eigenvalues and eigenvectors of a symmetric matrix, a Hermitian in the complex case. Symmetric eigenvalue server classes, which allow the computation of only the eigenvalues in a given range and (optionally) their corresponding eigenvectors. Symmetric eigenvalue server classes and the Hermitian eigenvalue server class, respectively, allow the computation of a subset of the eigenvalues and (optionally) their corresponding eigenvectors. www.roguewave.com /support/docs/sourcepro/lapackref/2-5.html   (210 words)

 The Algebraic Eigenvalue Problem The algebraic eigenvalue problem refers to finding a set of characteristic values associated with a matrix or matrices. Eigenvalues and eigenvectors are important in that when the corresponding equations model a physical situation they tell us useful information about it. Note that if B = I, the identity matrix, then this is the traditional eigenvalue problem. www.numerical-methods.com /eigen.htm   (178 words)

 15.13. Eigenvalue and Eigenvector Extraction   (Site not responding. Last check: 2007-10-15) The eigenvectors associated with multiple eigenvalues are evaluated using initial vector deflation by Gram-Schmidt orthogonalization in the inverse iteration procedure. The QR algorithm (Wilkinson(18)) is used to extract the eigenvalues of the [B] matrix. The 2n eigenvalues of Equation 15.168 are calculated by using the QR algorithm (Press et al.(254)). www.oulu.fi /atkk/tkpalv/unix/ansys-6.1/content/thy_tool13.html   (2420 words)

 Theory and Numerics of Matrix Eigenvalue Problems The aim of this workshop is to bring together leading researchers in the theory and numerical solution of eigenvalue problems with a view to surveying the state of the art, promoting collaboration, and making progress on the many challenging problems in this area. A unique feature of the workshop will be the chance for researchers from all parts of the spectrum from core linear algebra to numerical linear algebra to interact and work together intensively for the duration of the workshop. All other conferences in core and numerical linear algebra are much broader, and so do not allow the focus on eigenvalue problems or the bringing together of a critical mass of leading eigenvalue researchers in an environment in which they can work together. www.pims.math.ca /birs/workshops/2003/03w5008   (375 words)

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