scholarly journals INVERSE EIGENVALUE PROBLEM FOR EUCLIDEAN DISTANCE MATRICES OF SIZE 3

2012 ◽  
Vol 87 (1) ◽  
pp. 82-93 ◽  
Author(s):  
GAŠPER JAKLIČ ◽  
JOLANDA MODIC
Keyword(s):  
Eigenvalue Problem ◽  
Euclidean Distance ◽  
Distance Matrix ◽  
Inverse Eigenvalue Problem ◽  
Matrix Elements ◽  
Euclidean Distance Matrix ◽  
All Solutions ◽  
The Matrix ◽  
Inverse Eigenvalue ◽  
Euclidean Distance Matrices

AbstractA matrix is a Euclidean distance matrix (EDM) if there exist points such that the matrix elements are squares of distances between the corresponding points. The inverse eigenvalue problem (IEP) is as follows: construct (or prove the existence of) a matrix with particular properties and a given spectrum. It is well known that the IEP for EDMs of size 3 has a solution. In this paper all solutions of the problem are given and their relation with geometry is studied. A possible extension to larger EDMs is tackled.

scholarly journals A Test Matrix for an Inverse Eigenvalue Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
G. M. L. Gladwell ◽  
T. H. Jones ◽  
N. B. Willms
Keyword(s):  
Inverse Problem ◽  
Eigenvalue Problem ◽  
Explicit Solution ◽  
Additional Condition ◽  
Tridiagonal Matrix ◽  
Inverse Eigenvalue Problem ◽  
Test Matrix ◽  
Symmetric Tridiagonal Matrix ◽  
The Matrix ◽  
Inverse Eigenvalue

We present a real symmetric tridiagonal matrix of ordernwhose eigenvalues are{2k}k=0n-1which also satisfies the additional condition that its leading principle submatrix has a uniformly interlaced spectrum,{2l+1}l=0n-2. The matrix entries are explicit functions of the sizen, and so the matrix can be used as a test matrix for eigenproblems, both forward and inverse. An explicit solution of a spring-mass inverse problem incorporating the test matrix is provided.

The combinatorial inverse eigenvalue problem II: all cases for small graphs

2014 ◽  
Vol 27 ◽  
Author(s):  
Wayne Barrett ◽  
Curtis Nelson ◽  
John Sinkovic ◽  
Tianyi Yang
Keyword(s):  
Eigenvalue Problem ◽  
Undirected Graph ◽  
Complete Solution ◽  
Inverse Eigenvalue Problem ◽  
Pendant Vertex ◽  
Real Numbers ◽  
Connected Graphs ◽  
The Matrix ◽  
Inverse Eigenvalue

Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n by n matrices whose nonzero off-diagonal entries correspond to the edges of G. Given 2n - 1 real numbers \lambda_1\geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \cdots \geq \lambda_{n-1} \geq \mu_{n-1} \geq \lambda_{n-1}, and a vertex v of G, the question is addressed of whether or not there exists A in S(G) with eigenvalues \lambda_1, \ldots, \lambda_ n such that A(v) has eigenvalues \mu_1, \ldots, \mu_{n-1}, where A(v) denotes the matrix with vth row and column deleted. A complete solution can be given for the path on n vertices with v a pendant vertex and also for the star on n vertices with v the dominating vertex. The main result is a complete solution to this "\lambda, \mu" problem for all connected graphs on 4 vertices.

scholarly journals Numerical construction of structured matrices with given eigenvalues

2019 ◽  
Vol 7 (1) ◽  
pp. 263-271
Author(s):  
Brian D. Sutton
Keyword(s):  
Eigenvalue Problem ◽  
Constrained Optimization ◽  
Numerical Optimization ◽  
Optimization Problem ◽  
Symmetric Matrix ◽  
Inverse Eigenvalue Problem ◽  
Constrained Optimization Problem ◽  
The Matrix ◽  
Inverse Eigenvalue ◽  
Optimization Routine

Abstract We consider a structured inverse eigenvalue problem in which the eigenvalues of a real symmetric matrix are specified and selected entries may be constrained to take specific numerical values or to be nonzero. This includes the problem of specifying the graph of the matrix, which is determined by the locations of zero and nonzero entries. In this article, we develop a numerical method for constructing a solution to the structured inverse eigenvalue problem. The problem is recast as a constrained optimization problem over the orthogonal manifold, and a numerical optimization routine seeks its solution.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

scholarly journals Ad hoc microphone array calibration: Euclidean distance matrix completion algorithm and theoretical guarantees

2015 ◽  
Vol 107 ◽  
pp. 123-140 ◽  
Author(s):  
Mohammad J. Taghizadeh ◽  
Reza Parhizkar ◽  
Philip N. Garner ◽  
Hervé Bourlard ◽  
Afsaneh Asaei
Keyword(s):  
Euclidean Distance ◽  
Ad Hoc ◽  
Microphone Array ◽  
Matrix Completion ◽  
Distance Matrix ◽  
Euclidean Distance Matrix ◽  
Array Calibration ◽  
Euclidean Distance Matrix Completion
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