Eigenvalue problem for an irregular λ-matrix

The solution of the eigenvalue problem is examined for the polyomial matrixD()=A_o^s+A₁^s–1+...+A_s when the matricesA ₀ andA ₂ (or one of them) are singular. A normalized process is used for solving the problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigenvalue of matrixD() and to the zero eigenvalue of matrixA ₀. The computation of the other eigenvalues ofD() is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 80–92, 1976.     

Pólya convertibility problem for symmetric matrices

We investigate symmetric (0, 1) matrices on which the permanent is convertible to the determinant by assigning ± signs to their entries. In particular, we obtain several quantitative bounds for the number of nonzero elements of such matrices, including the analog of Gibson’s theorem for symmetric matrices.     

A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem

In this paper, we investigate a DC (Difference of Convex functions) programming technique for solving large scale Eigenvalue Complementarity Problems (EiCP) with real symmetric matrices. Three equivalent formulations of EiCP are considered. We first reformulate them as DC programs and then use DCA (DC Algorithm) for their solution. Computational results show the robustness, efficiency, and high speed of the proposed algorithms.     

Eigenvalue inequalities for the clamped plate problem of the ■ operator

The ■ operator is introduced by Xin(2015), which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend the ■ operator to a more general elliptic differential operator ■, and investigate the clamped plate problem of the bi-■ operator,which is denoted by ■ on the complete Riemannian manifolds. A general formula of eigenvalues for the ■ operator is established. Applying this formula, we estimate the eigenvalues on th...     

A geometrical method for diagonalizing real,symmetric 3×3 matrices through Euler rotations

The classical problem of finding the principal values (eigenvalues) and principal axes (eigenvectors) of a physical property represented by a second-rank symmetric tensor is treated in textbooks by solving the characteristic equation associated with the 3×3 symmetric matrix representation. The same problem is solved here without reference to the characteristic equation. By use of Euler rotations, analytical expressions are attained for the Euler eigenangles, the eigenvalues and the eigenvectors.     

Relative invariants of 3 × 3 matrix triples ∗

We present primary and secondary generators for the algebra of polynomial invariants of the direct product of two copies of the special linear group Sl ₃ acting naturally on triples of 3 × 3 matrices over a field of characteristic zero. We handle also the analogous problem for triples and quadruples of 2 × 2 matrices.     

Spectral analysis of the preconditioned system for the 3 × 3 block saddle point problem

Numerical Algorithms - In this work, we consider some preconditioning techniques for a class of 3 × 3 block saddle point problems, which arise from finite element methods for solving...     

A symmetric BEM–FEM method for an axisymmetric eddy current problem

This paper deals with the numerical solution of the time-harmonic eddy current model in an axisymmetric unbounded domain. To this end, a new symmetric BEM–FEM formulation is derived and also analyzed. Moreover, error estimates for the corresponding discretization are proven.     

On the isospectral orbifold–manifold problem for nonpositively curved locally symmetric spaces

An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel’s conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.     

Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy

The paper studies the global convergence of the Jacobi method for symmetric matrices of size 4. We prove global convergence for all 720 cyclic pivot strategies. Precisely, we show that inequality S(A ^[t+3]) ≤ γ S(A ^[t]), t ≥ 1, holds with the constant γ < 1 that depends neither on the matrix A nor on the pivot strategy. Here, A ^[t] stands for the matrix obtained from A after t full cycles of the Jacobi method and S(A) is the off-diagonal norm of A. We show why three consecutive cycles have to be considered. The result has a direct application on the J-Jacobi method.     

The Kadison–Singer problem for the direct sum of matrix algebras

Let M _n denote the algebra of complex n × n matrices and write M for the direct sum of the M _n . So a typical element of M has the form

On some constructive methods for the matrix Riemann–Hilbert boundary-value problem

In this paper we consider the relations between the Riemann–Hilbert monodromy problem and the matrix Riemann–Hilbert boundary-value problem with piecewise continuous coefficient and construct the so-called canonical matrix for the boundary-value problem for a piecewise continuous matrix-function. The formula for the calculation of the index is also obtained.     

Invertible symmetric 3 × 3 binary matrices and GQ(2, 4)

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3?×?3 matrices over GF(2) and the points of the generalized quadrangle GQ(2,?4). The 15 matrices with eigenvalue one correspond to a copy of the subquadrangle GQ(2,?2), whereas the 12 matrices without eigenvalues have their geometric counterpart in the associated double-six. The fine details of this correspondence, including the precise algebraic meaning/analogue of collinearity, are furnished by employing the representation of GQ(2,?4) as a quadric in PG(5,?2) of projective index one. An interesting physics application of our findings is also mentioned.     

A Two-Level Method for Nonsymmetric Eigenvalue Problems

A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.     

Existence of triple symmetric positive solutions for four-point boundary-value problem with one-dimensional p-Laplacian

In this paper, we consider the four-point boundary value problem for one-dimensional p-Laplacian $$\bigl(\phi_{p}(u'(t))\bigr)'+q(t)f\bigl(t,u(t),u'(t)\bigr)=0,\quad t\in(0,1),$$ subject to the boundary conditions $$u(0)-\beta u'(\xi)=0,\qquad u(1)+\beta u'(\eta)=0,$$ where φ _p (s)=|s| ^p?2 s. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three symmetric positive solutions to the above boundary value problem. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly.     

Generalized factorization for a class of non-rational 2×2 matrix functions

In this paper the generalized factorization for a class of 2×2 piecewise continuous matrix functions on is studied. Using a space transformation the problem is reduced to the generalized factorization of a scalar piecewise continuous function on a contour in the complex plane. Both canonical and non-canonical generalized factorization of the original matrix function are studied.Sponsored by J.N.I.C.T. (Portugal) under grant no. 87422/MATM