Eigenvalue Problem For Matrices Research Articles

The inverse eigenvalue problem of structured matrices from the design of Hopfield neural networks

By means of the properties of structured matrices from the design of Hopfield neural networks, we establish the necessary and sufficient conditions for the solvability of the inverse eigenvalue problem AX=XΛ in structured matrix set SARJn. In the case where AX=XΛ is solvable in SARJn, we derive the generalized representation of the solutions. In addition, in corresponding solution set of the equation, we provide the explicit expression of the nearest matrix to a given matrix in the Frobenius norm.

Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in O(n) operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from N. Jakovčević Stor et al. (2015) [16].

Eigenvalue problem of Hamiltonian operator matrices

This paper deals with the eigenvalue problem of Hamiltonian operator matrices with at least one invertible off-diagonal entry. The ascent and the algebraic multiplicity of their eigenvalues are determined by using the properties of the eigenvalues and associated eigenvectors. The necessary and sufficient condition is further given for the eigenvector (root vector) system to be complete in the Cauchy principal value sense.

Avoiding Communication in Successive Band Reduction

The running time of an algorithm depends on both arithmetic and communication (i.e., data movement) costs, and the relative costs of communication are growing over time. In this work, we present sequential and distributed-memory parallel algorithms for tridiagonalizing full symmetric and symmetric band matrices that asymptotically reduce communication compared to previous approaches. The tridiagonalization of a symmetric band matrix is a key kernel in solving the symmetric eigenvalue problem for both full and band matrices. In order to preserve structure, tridiagonalization routines use annihilate-and-chase procedures that previously have suffered from poor data locality and high parallel latency cost. We improve both by reorganizing the computation and obtain asymptotic improvements. We also propose new algorithms for reducing a full symmetric matrix to band form in a communication-efficient manner. In this article, we consider the cases of computing eigenvalues only and of computing eigenvalues and all eigenvectors.

An algorithm for constructing nonnegative matrices with prescribed real eigenvalues

Provided with the real spectrum, this paper presents a numerical procedure based on the induction principle to solve two types of inverse eigenvalue problems, one for nonnegative matrices and another for symmetric nonnegative matrices. As an immediate application, our approach can offer not only the sufficient condition to solve inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices, but also a numerical way to solve inverse eigenvalue problems for stochastic matrices. Numerical examples are presented to show the capacity of our method.

A finite-step construction of totally nonnegative matrices with specified eigenvalues

Matrices where all minors are nonnegative are said to be totally nonnegative (TN) matrices. In the case of banded TN matrices, which can be expressed by products of several bidiagonal TN matrices, Fukuda et al. (Annal. Mat. Pura Appl. 192, 423---445, 2013) discussed the eigenvalue problem from the viewpoint of the discrete hungry Toda (dhToda) equation. The dhToda equation is a discrete integrable system associated with box and ball systems. In this paper, we consider an inverse eigenvalue problem for such banded TN matrices by examining the properties of the dhToda equation. This problem is a real-valued nonnegative inverse eigenvalue problem. First, we show the determinant solution to the dhToda equation with suitable boundary conditions. Next, we clarify the relationship between the characteristic polynomials of the banded TN matrices and the determinant solution. Finally, taking this relationship into account, we design a finite-step procedure for constructing banded TN matrices with specified eigenvalues. We also present an example to demonstrate this procedure.

Barzilai-Borwein-like methods for the extreme eigenvalue problem

We consider numerical methods for the extreme eigenvalue problem of large scale symmetric positive definite matrices. By the variational principle, the extreme eigenvalue can be obtained by minimizing some unconstrained optimization problem. Firstly, we propose two adaptive nonmonotone Barzilai-Borwein-like methods for the unconstrained optimization problem. Secondly, we prove the global convergence of the two algorithms under some conditions. Thirdly, we compare our methods with eigsand the power method for the standard test problems from the UF Sparse Matrix Collection. The primary numerical experiments indicate that the two algorithms are promising.

Generalized inverse eigenvalue problem for matrices whose graph is a path

In this paper, we analyse a special generalized inverse eigenvalue problem Anx=λBnx for the pair (An,Bn) of matrices each of whose graph is a path on n vertices, by investigating the leading principal minors of the matrix An−λBn. From the given data consisting of Bn, a matrix Ak (k<n) whose graph is a path on k vertices, two column vectors X2,Y2∈Rn−k and distinct real numbers λ and μ we construct An and two column vectors X1,Y1∈Rk such that Ak is the leading principal sub-matrix of An and (λ,X), (μ,Y) are the eigenpairs of (An,Bn), where X=(X1T,X2T)T and Y=(Y1T,Y2T)T. Further, numerical examples are also given to demonstrate the applicability of the results developed here.

Linear parameterized inverse eigenvalue problem of bisymmetric matrices

In this paper, we consider the linear parameterized inverse eigenvalue problem of bisymmetric matrices which is described as follows:Problem IGiven real bisymmetric matrices B0,B1,B2,…,Bn, and real numbers λ1⁎,λ2⁎,…,λn⁎, find values of c=(c1,c2,…,cn)T such that the eigenvalues of the matrixB(c)=B0+c1B1+c2B2+⋯+cnBn are precisely λ1⁎,λ2⁎,…,λn⁎.We first establish the structure of bisymmetric matrices. Based on the Newton-type iterative method, we then turn Problem I into the inverse eigenvalue problems of symmetric matrices with small sizes, and solve these problems. Finally, we present numerical examples to illustrate our results.

A CONTRIBUTION TO THE INVERSE EIGENVALUE PROBLEM FOR NON-NEGATIVE MATRICES

Sometime in the early 1970s Trevor introduced me to the spectral theory of positive linear operators which owes its origins to the celebrated Perron-Frobenius theorem according to which the spectral radius of a non-negative matrix is one of its eigenvalues, and possesses a corresponding eigenvector whose components are nonnegative real numbers. This was a subject dear to his heart, and a recurring theme to which he often returned in later years. In this article we make a tangential contribution to the converse of the Perron-Frobenius theorem, the so-called inverse eigenvalue problem for non-negative matrices, namely, under what circumstances are the components of a vector of complex numbers the eigenvalues of such a matrix? To this end, we associate with each vector of unit norm an analytic self map of the unit open disc of the complex plane, which is also a rational function, and develop its power series expansion about the origin. Sufficient conditions are presented that ensure that the resulting coefficients—which encode information about the chosen vector—are non-negative. Conversely, if these are all non-negative, it turns out that the vector satisfies conditions that are necessary ones for it to solve the inverse problem.

Designing Various Multivariate Analysis at Will via Generalized Pairwise Expression

It is well known that dimensionality reduction based on multivariate analysis methods and their kernelized extensions can be formulated as generalized eigenvalue problems of scatter matrices, Gram matrices or their aug- mented matrices. This paper provides a generic and theoretical framework of multivariate analysis introducing a new expression for scatter matrices and Gram matrices, called Generalized Pairwise Expression (GPE). This expression is quite compact but highly powerful. The framework includes not only (1) the traditional multivariate analysis methods but also (2) several regularization techniques, (3) localization techniques, (4) clustering methods based on generalized eigenvalue problems, and (5) their semi-supervised extensions. This paper also presents a methodology for designing a desired multivariate analysis method from the proposed framework. The methodology is quite simple: adopting the above mentioned special cases as templates, and generating a new method by combining these templates appropriately. Through this methodology, we can freely design various tailor-made methods for specific purposes or domains.

A Contribution to the Inverse Eigenvalue Problem for Non-Negative Matrices

Sometime in the early 1970s Trevor introduced me to the spectral theory of positive linear operators which owes its origins to the celebrated Perron-Frobenius theorem according to which the spectral radius of a non-negative matrix is one of its eigenvalues, and possesses a corresponding eigenvector whose components are nonnegative real numbers. This was a subject dear to his heart, and a recurring theme to which he often returned in later years. In this article we make a tangential contribution to the converse of the PerronFrobenius theorem, the so-called inverse eigenvalue problem for non-negative matrices, namely, under what circumstances are the components of a vector of complex numbers the eigenvalues of such a matrix? To this end, we associate with each vector of unit norm an analytic self map of the unit open disc of the complex plane, which is also a rational function, and develop its power series expansion about the origin. Sufficient conditions are presented that ensure that the resulting coefficients which encode information about the chosen vector are non-negative. Conversely, if these are all non-negative, it turns out that the vector satisfies conditions that are necessary ones for it to solve the inverse problem.

A new structure-preserving method for quaternion Hermitian eigenvalue problems

In this paper we propose a novel structure-preserving algorithm for solving the right eigenvalue problem of quaternion Hermitian matrices. The algorithm is based on the structure-preserving tridiagonalization of the real counterpart for quaternion Hermitian matrices by applying orthogonal JRS-symplectic matrices. The algorithm is numerically stable because we use orthogonal transformations; the algorithm is very efficient, it costs about a quarter arithmetical operations, and a quarter to one-eighth CPU times, comparing with standard general-purpose algorithms. Numerical experiments are provided to demonstrate the efficiency of the structure-preserving algorithm.

A note on “Methods for constructing distance matrices and the inverse eigenvalue problem”

In this paper, a symmetric nonnegative matrix with zero diagonal and given spectrum, where exactly one of the eigenvalues is positive, is constructed. This solves the symmetric nonnegative eigenvalue problem (SNIEP) for such a spectrum. The construction is based on the idea from the paper Hayden, Reams, Wells, “Methods for constructing distance matrices and the inverse eigenvalue problem”. Some results of this paper are enhanced. The construction is applied for the solution of the inverse eigenvalue problem for Euclidean distance matrices, under some assumptions on the eigenvalues.

To solving inverse eigenvalue problems for matrices

The paper considers additive and multiplicative inverse eigenvalve problems and suggests a solution method based on the connection of inverse problems with the problem of computing the regular spectrum of a multiparameter pencil of constant matrices corresponding to the problem considered. Necessary and sufficient conditions for the solvability of inverse eigenvalve problems are established. Bibliography: 3 titles.

To solving the eigenvalue problem for polynomial matrices of general form

The paper considers the eigenvalve problem for a polynomial m × n matrix F(μ) of rank ρ. Algorithms. allowing one to reduce this problem to the generalized matrix eigenvalve problem are suggested. The algorithms are based on combining rank factorization methods with the method of hereditary pencils. Methods for exhausting the subspaces of polynomial solutions of zero index from the matrix nullspaces and for isolating the regular kernel from F(μ), with its subsequent linearization, are proposed. Bibliography: 6 titles.

Communication avoiding successive band reduction

The running time of an algorithm depends on both arithmetic and communication (i.e., data movement) costs, and the relative costs of communication are growing over time. In this work, we present both theoretical and practical results for tridiagonalizing a symmetric band matrix: we present an algorithm that asymptotically reduces communication, and we show that it indeed performs well in practice. The tridiagonalization of a symmetric band matrix is a key kernel in solving the symmetric eigenvalue problem for both full and band matrices. In order to preserve sparsity, tridiagonalization routines use annihilate-and-chase procedures that previously have suffered from poor data locality. We improve data locality by reorganizing the computation, asymptotically reducing communication costs compared to existing algorithms. Our sequential implementation demonstrates that avoiding communication improves runtime even at the expense of extra arithmetic: we observe a 2x speedup over Intel MKL while doing 43% more floating point operations. Our parallel implementation targets shared-memory multicore platforms. It uses pipelined parallelism and a static scheduler while retaining the locality properties of the sequential algorithm. Due to lightweight synchronization and effective data reuse, we see 9.5x scaling over our serial code and up to 6x speedup over the PLASMA library, comparing parallel performance on a ten-core processor.

On the inverse eigenvalue problem for nonnegative matrices of order two to five

In this paper, for a given set of numbers with special conditions, we construct a nonnegative matrix such that the given set is its spectrum.

Parallel Rayleigh Quotient Optimization with FSAI‐Based Preconditioning

The present paper describes a parallel preconditioned algorithm for the solution of partial eigenvalue problems for large sparse symmetric matrices, on parallel computers. Namely, we consider the Deflation‐Accelerated Conjugate Gradient (DACG) algorithm accelerated by factorized‐sparse‐approximate‐inverse‐ (FSAI‐) type preconditioners. We present an enhanced parallel implementation of the FSAI preconditioner and make use of the recently developed Block FSAI‐IC preconditioner, which combines the FSAI and the Block Jacobi‐IC preconditioners. Results onto matrices of large size arising from finite element discretization of geomechanical models reveal that DACG accelerated by these type of preconditioners is competitive with respect to the available public parallel hypre package, especially in the computation of a few of the leftmost eigenpairs. The parallel DACG code accelerated by FSAI is written in MPI‐Fortran 90 language and exhibits good scalability up to one thousand processors.

Large a quasi-Jacobi form for J-normal matrices and inverse eigenvalue problems

A quasi-Jacobi form for J-unitarily diagonalizable J-normal matrices is given, extending a result for normal matrices due to Malamud. The inverse eigenvalue problem for J-normal matrices satisfying certain prescribed spectral conditions is investigated. It is shown that there exists unicity in the case of pseudo-Jacobi matrices.