The Eigenvalue Problem For Symmetric Matrices

1990 ◽  
pp. 390-439
Author(s):  
Heinz Rutishauser
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrices

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

scholarly journals Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

2019 ◽  
Vol 7 (1) ◽  
pp. 276-290
Author(s):  
Mohammad Adm ◽  
Shaun Fallat ◽  
Karen Meagher ◽  
Shahla Nasserasr ◽  
Sarah Plosker ◽  
...  
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Inverse Eigenvalue ◽  
Complete Multipartite

Abstract Associated to a graph G is a set 𝒮(G) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in 𝒮 (G) partition n; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in 𝒮 (G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in 𝒮 (G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in 𝒮 (G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.

scholarly journals Explicit Solution of the Inverse Eigenvalue Problem of Real Symmetric Matrices and Its Application to Electrical Network Synthesis

2008 ◽  
Vol 2008 ◽  
pp. 1-25 ◽  
Author(s):  
D. B. Kandić ◽  
B. D. Reljin
Keyword(s):  
Eigenvalue Problem ◽  
Common Ground ◽  
Explicit Construction ◽  
Inverse Eigenvalue Problem ◽  
Electrical Network ◽  
Symmetric Matrices ◽  
Network Synthesis ◽  
Sign Patterns ◽  
Specific Sign ◽  
Inverse Eigenvalue

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving-point immittance functions of transformerless, common-ground, two-element-kindRLCnetworks and in generation of their equivalent realizations.

PARALLEL SUBSPACE ITERATION METHOD FOR THE SPARSE SYMMETRIC EIGENVALUE PROBLEM

2006 ◽  
Vol 05 (04) ◽  
pp. 801-818 ◽  
Author(s):  
RICHARD LOMBARDINI ◽  
BILL POIRIER
Keyword(s):  
Eigenvalue Problem ◽  
Iteration Method ◽  
Degrees Of Freedom ◽  
Sparse Matrices ◽  
Model Systems ◽  
Symmetric Matrices ◽  
Six Degrees Of Freedom ◽  
Subspace Iteration ◽  
The Matrix ◽  
Parallel Iterative

A new parallel iterative algorithm for the diagonalization of real sparse symmetric matrices is introduced, which uses a modified subspace iteration method. A novel feature is the preprocessing of the matrix prior to iteration, which allows for a natural parallelization resulting in a great speedup and scalability of the method with respect to the number of compute nodes. The method is applied to Hamiltonian matrices of model systems up to six degrees of freedom, represented in a truncated Weyl–Heisenberg wavelet (or "weylet") basis developed by one of the authors (Poirier). It is shown to accurately determine many thousands of eigenvalues for sparse matrices of the order N ≈ 106, though much larger matrices may also be considered.

scholarly journals Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems

1991 ◽  
Vol 32 (4) ◽  
pp. 437-456 ◽  
Author(s):  
D. C. Dzeng ◽  
W. W. Lin
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Positive Semidefinite ◽  
Symmetric Matrices ◽  
Homotopy Continuation ◽  
Smooth Curves ◽  
Homotopy Continuation Method ◽  
Symmetric Eigenvalue Problems

AbstractWe consider a generalised symmetric eigenvalue problem Ax = λMx, where A and M are real n by n symmetric matrices such that M is positive semidefinite. The purpose of this paper is to develop an algorithm based on the homotopy methods in [9, 11] to compute all eigenpairs, or a specified number of eigenvalues, in any part of the spectrum of the eigenvalue problem Ax = λMx. We obtain a special Kronecker structure of the pencil A − λM, and give an algorithm to compute the number of eigenvalues in a prescribed interval. With this information, we can locate the lost eigenpair by using the homotopy algorithm when multiple arrivals occur. The homotopy maintains the structures of the matrices A and M (if any), and the homotopy curves are n disjoint smooth curves. This method can be used to find all/some isolated eigenpairs for large sparse A and M on SIMD machines.

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