On the Deligne-Simpson problem and its weak version

We consider the Deligne-Simpson problem (DSP) (respectively the weak DSP): Give necessary and sufficient conditions upon the choice of the p+1 conjugacy classes or so that there exist irreducible (p+1)-tuples (respectively (p+1)-tuples with trivial centralizers) of matrices A_j∈c_j with zero sum or of matrices M_j∈C_j whose product is I. The matrices A_j (respectively M_j) are interpreted as matrices-residua of Fuchsian linear systems (respectively as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues.     

Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree

Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree we focus upon M₂, the maximum value of the sum of the two largest multiplicities. The corresponding M₁ is already understood. The notion of assignment (of eigenvalues to subtrees) is formalized and applied. Using these ideas, simple upper and lower bounds are given for M₂ (in terms of simple graph theoretic parameters), cases of equality are indicated, and a combinatorial algorithm is given to compute M₂ precisely. In the process, several techniques are developed that likely have more general uses.     

Positive powers of certain conditionally negative definite matrices

We investigate the spectrum of matrices (∣x_i,−x_j∣^a)ⁿ_i,j=1 with α>0 and distinct x₁,…,x_n whichare relevant to the theory of scattered data interpolation and spline functions. The main result is the non-singularity of these matrices, which is based on the property that the number of negativeand positive eigenvalues of these matrices is independent of x₁,…,x_n. Oscillation properties of asubset of eigenvectors of these matrices are also obtained. For 2<α<4 and points x¹,…,xⁿ∈R²,a sufficient condition for the non-singularity of (∥xⁱ−x^j∥^α₂)ⁿ_i,j=1 is derived.     

A matrix pencil approach computing the elementary divisors of a matrix: Numerical aspects and applications

In the present paper is presented a new matrix pencil-based numerical approach achieving the computation of the elementary divisors of a given matrixA ∈ C ^{n × n}. This computation is attained without performing similarity transformations and the whole procedure is based on the construction of the Piecewise Arithmetic Progression Sequence (PAPS) of the associated pencil λI _n - A of matrix A, for all the appropriate values of λ belonging to the set of eigenvalues of A. This technique produces a stable and accurate numerical algorithm working satisfactorily for matrices with a well defined eigenstructure. The whole technique can be applied for the computation of the first, second and Jordan canonical form of a given matrixA ∈ C ^{n × n}. The results are accurate for matrices possessing a well defined canonical form. In case of defective matrices, indications of the most appropriately computed canonical form are given.     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.     

Random evolutions in discrete and continuous time

This paper points out a connection between random evolutions and products of random matrices. This connection is useful in predicting the long-run growth rate of a single-type, continuously changing population in randomly varying environments using only observations at discrete points in time. A scalar Markov random evolution is specified by the n×n irreducible intensity matrix or infinitesimal generator Q = (q_ij) of a time-homogeneous Markov chain and by n finite real growth rates (scalars) s_i. The scalar Markov random evolution is the quantity M^C(t) = exp(Σⁿ_j=1s_jg^C_j (t)), where g^C_j(t) is the occupancy times in state j up to time t. The scalar Markov product of random matrices induced by this scalar Markov random evolution is the quantity M^D(t) = exp(Σⁿ_j=1s_jg^D_j (t)), where g^D_j(t) is the occupancy time in state j up to and including t of the discrete-time Markov chain with stochastic one-step transition matrix P = e^Q. We show that lim_t→∞(1/t)E(logM^D(t))=lim_t→∞(1/t)E(logM^C(t)) but that in general lim_t→∞(1/t)logE(M^C(t)) ≠ lim_t→∞(1/t)logE(M^D(t)). Thus the mean Malthusian parameter of population biologists is invariant with respect to the choice of continuous or discrete time, but the rate of growth of average population size is not. By contrast with a single-type population, in multitype populations whose growth is governed by non-commuting operators, the mean Malthusian parameter may be destined for a less prominent role as a measure of long-run growth.     

Simultaneous congruence of convex compact sets of Hermitian matrices with constant rank

Let S be a compact convex set of n × n hermitian matrices (n ⩾ 2). Suppose every member of S is nonsingular and has exactly one negative eigenvalue. Let (ε₁,…,ε_n) be any ordered n-tuple from the set {- 1, 1}. One of our main results is that a nonsingular matrix X exists such that, for every A in S and every 1 ⩽ j ⩽ n, the (j, j) entry of X¹AX has sign ε_j. A similar result, with only negative ε_j allowed, is proved also for a compact convex set S of n × n hermitian matrices such that every member of S has the same rank and exactly one negative eigenvalue.     

A multiscale Galerkin method for the hypersingular integral equation reduced by the harmonic equation

The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot(θ/2). Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j+1 × 2j+1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.     

The centro-invertible matrix: A new type of matrix arising in pseudo-random number generation

This paper defines a new type of matrix (which will be called a centro-invertible matrix) with the property that the inverse can be found by simply rotating all the elements of the matrix through 180 degrees about the mid-point of the matrix. Centro-invertible matrices have been demonstrated in a previous paper to arise in the analysis of a particular algorithm used for the generation of uniformly-distributed pseudo-random numbers.An involutory matrix is one for which the square of the matrix is equal to the identity. It is shown that there is a one-to-one correspondence between the centro-invertible matrices and the involutory matrices. When working in modular arithmetic this result allows all possible k by k centro-invertible matrices with integer entries modulo M to be enumerated by drawing on existing theoretical results for involutory matrices.Consider the k by k matrices over the integers modulo M. If M takes any specified finite integer value greater than or equal to two then there are only a finite number of such matrices and it is valid to consider the likelihood of such a matrix arising by chance. It is possible to derive both exact expressions and order-of-magnitude estimates for the number of k by k centro-invertible matrices that exist over the integers modulo M. It is shown that order (√N) of the N=M^(k2) different k by k matrices modulo M are centro-invertible, so that the proportion of these matrices that are centro-invertible is order (1/√N).     

Author index to volume 20

This paper considers the conjecture that given a real nonsingular matrix A, there exist a real diagonal matrix Λ with $\text{¦λ}{}_{\mathrm{ii}}\text{λ = 1}$ and a permutation matrix P such that (ΛPA) is positive stable. The conjecture is shown to be true for matrices of order 3 or less and may not be true for higher order matrices. A counterexample is presented in terms of a matrix of order 65. In showing this, an interesting matrix M_l of order 2^l = 64, which satisfies the matrix equation 2^l-1(M_l + M^T_l), has been used. The stability analysis is done by first decomposing the nonsingular matrix into its polar form. Some interesting results are presented in the study of eigenvalues of a product of orthogonal matrices. A simple function is derived in terms of these orthogonal matrices, which traces a hysteresis loop.     

A theorem on products of matrices

A new result on products of matrices is proved in the following theorem: let M_i (i=1,2,…) be a bounded sequence of square matrices, and K be the l.u.b. of the spectral radii ρ(M_i). Then for any positive number ε there is a constant A and an ordering p(j) (j = 1,2,…) of the matrices such that

$\text{∏}\text{j=1}\text{n}\text{M}{}_{\mathrm{p(j)}}\text{?A·(K+ε)}{}^{\mathrm{n}}\text{(n = 1,2,…)}$

. The ordering is well defined by p(j), a one-to-one mapping on the set of positive integers. In general the inequality does not hold for any ordering p(j) (a counterexample is provided); however, some sufficient conditions are given for the result to remain true irrespective of the order of the matrices.     

Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices

The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX ? XB = C and X ? AXB = C can be transformed to equations whose coefficients are block companion matrices: $\text{C}\text{?}{}_{\mathrm{L}}\text{X?XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$ and $\text{X?}\text{C}\text{?}{}_{\mathrm{L}}\text{XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$, respectively, where ?_L and C_M stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λ^sI + Σ^s?1_j=0λ^jL_j and M(λ) = λ^tI + Σ^t7minus;1_j=0λ^jM_j. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ?, where the matrix S is r²l × r²l[l = max(s, t)] and enjoys some symmetry properties.     

Non-induced isomorphisms of matrix rings

We give examples of distinct integersi, j, and ringsT for which the matrix ringsM _i (T) andM _j (T) are isomorphic as rings, but for which the free modules _T T ⁽ⁱ⁾ and _T T ⁽ⁱ⁾ are non-isomorphic asT-modules.     

Idempotent elements determined matrix algebras

Let M_n(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from M_n(R)×M_n(R) to V satisfies the condition that {u,u}={e,u} whenever u²=u, then there exists a linear map f from M_n(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on M_n(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M_n(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.     

Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes,a new preprocessing approach,called multineighboring grids(MNG),is proposed for solving PDE eigen-problems on an existing grid G(Δ).The linear or multi-linear element,based on box-splines,are taken as the frst stage Kh1Uh=λh1Mh1Uh.In this paper,the j-th stage neighboring-grid scheme is defned asKh jUh=λh j Mh jUh,where Kh j:=Mh j 1Kh1and Mh jUh is to be found as a better mass distribution over the j-th stage neighboring-gridG(Δ),and Kh jcan be seen as an expansion of Kh1on the j-th neighboring-grid with respect to the(j 1)-th mass distribution Mh j 1.It is shown that for an ODE model eigen-problem,the j-th stage scheme with 2j-th order B-spline basis can reach2j-th order accuracy and even(2j+2)-th order accuracy by perturbing the mass matrix.The argument can be extended to high dimensions with separable variable cases.For Laplace eigen-problems with some 2-D and 3-D structured uniform grids,some 2j-th order schemes are presented for j 3.     

Eigenvector distribution of Wigner matrices

We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν _ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν _ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.     

Laguerre polynomial expansions in indefinite inner product spaces

When ?j ? 1 < α < ?j, where j is a positive integer, the Laguerre polynomials {L_n^(α)}_{n = 0}^∞ form a complete orthogonal set in a nondegenerate inner product space H which is defined by employing an appropriate regularized linear functional on H^(j)[[0, ∞); x^{α + j}e^?x]. Expansions in terms of these Laguerre polynomials are exhibited. The Laguerre differential operator is shown to be self-adjoint with real, discrete, integer eigenvalues. Its spectral resolution and resolvent are exhibited and discussed.     

Invariants of matrix pairs over discrete valuation rings and Littlewood-Richardson fillings

Let M and N be two r×r matrices of full rank over a discrete valuation ring R with residue field of characteristic zero. Let P,Q and T be invertible r×r matrices over R. It is shown that the orbit of the pair (M,N) under the action (M,N)?(PMQ^-1,QNT^-1) possesses a discrete invariant in the form of Littlewood-Richardson fillings of the skew shape λ/μ with content ν, where μ is the partition of orders of invariant factors of M, ν is the partition associated to N, and λ the partition of the product MN. That is, we may interpret Littlewood-Richardson fillings as a natural invariant of matrix pairs. This result generalizes invariant factors of a single matrix under equivalence, and is a converse of the construction in Appleby (1999) [1], where Littlewood-Richardson fillings were used to construct matrices with prescribed invariants. We also construct an example, however, of two matrix pairs that are not equivalent but still have the same Littlewood-Richardson filling. The filling associated to an orbit is determined by special quotients of determinants of a matrix in the orbit of the pair.     

On certain spaces of lattice diagram determinants

The aim of this work is to study some lattice diagram determinants Δ_L(X,Y) as defined in (Adv. Math. 142 (1999) 244) and to extend results of Aval et al. (J. Combin. Theory Ser. A, to appear). We recall that M_L denotes the space of all partial derivatives of Δ_L. In this paper, we want to study the space M_i,j^k(X,Y) which is defined as the sum of M_L spaces where the lattice diagrams L are obtained by removing k cells from a given partition, these cells being in the “shadow” of a given cell (i,j) in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space M_i,j^k(X,Y), that we conjecture to be optimal. This dimension is a multiple of n! and thus we obtain a generalization of the n! conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the “shift” operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace M_i,j^k(X) consisting of elements of 0 Y-degree.