Convergence of quasi-Newton matrices generated by the symmetric rank one update

Quasi-Newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives. This paper gives conditions under which these approximations can be proved to converge globally to the true Hessian matrix, in the case where the Symmetric Rank One update formula is used. The rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates. The theory is confirmed by some numerical experiments that also show the convergence of the Hessian approximations to be substantially slower for other known quasi-Newton formulae.The work of this author was supported by the National Sciences and Engineering Research Council of Canada, and by the Information Technology Research Centre, which is funded by the Province of Ontario.     

Stochastic matrices over cones

The notion of a stochastic operator in an ordered Banach space is specialized to a finite dimensional ordered real vector space. The classical limit theorems are obtained, and an application is made to non-homogeneous Markov chains. Finally, groups of nonnegative matrices are discussed.     

Stochastic matrices over cones

The notion of a stochastic operator in an ordered Banach space is specialized to a finite dimensional ordered real vector space. The classical limit theorems are obtained, and an application is made to non-homogeneous Markov chains. Finally, groups of nonnegative matrices are discussed.     

Combining dissimilarity matrices by using rank correlations

Recently there has been increased interest in methods for aggregating multiple matrices observed on a fixed set of entities, where each matrix expresses a particular notion of the dissimilarity of one entity from another. An optimization-based procedure is developed, which returns a global dissimilarity matrix in the form of a weighted average of the partial matrices. The weights are determined with the goal of overcoming conflicts and overlaps that inevitably arise when different sources of data become part of the same representational structure. One important aspect in this context is the coefficient used to measure the degree of association between matrices. Here, it is normal to adopt the vector correlation proposed by Escoufier, but this has the drawback of depending on the Pearson’s correlation, which is highly prone to the effects of outliers. The solution that we propose to mitigate this problem is the substitution of the Pearson coefficient with rank correlations that are less affected by errors of measurement, nonlinearity or outliers. The results obtained with real and simulated data confirm that applying vector rank correlations attenuates the adverse effects of anomalies and, in the case of clean and faultless data, yields weights which basically conform to those obtained using the Escoufier coefficient.     

Triangular stochastic matrices generated by infinitesimal elements

We show that each element in the semigroup S _n of all n × n non-singular upper (or lower) triangular stochastic matrices is generated by the infinitesimal elements of S _n, which form a cone consisting of all n × n upper (or lower) triangular intensity matrices.     

Semipositive matrices and their semipositive cones

The semipositive cone of \(A\in \mathbb {R}^{m\times n}, K_A = \{x\ge 0\,:\, Ax\ge 0\}\), is considered mainly under the assumption that for some \(x\in K_A, Ax>0\), namely, that A is a semipositive matrix. The duality of \(K_A\) is studied and it is shown that \(K_A\) is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.

     

Minimum rank of skew-symmetric matrices described by a graph

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem.     

Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X₁,…,X_t) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X₁,…,X_t with constant term 0 such that for some μK occurring in the coefficients of f(X₁,…,X_t). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x₁,…,x_t) is bounded above by a same natural number for all x₁,…,x_tρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x₁,…,x_t) for x₁,…,x_tρ.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Ordered partitions and codes generated by circulant matrices

We consider the set of ordered partitions of n into m parts acted upon by the cyclic permutation (12…m). The resulting family of orbits $\text{P}\text{(n, m)}$ is shown to have cardinality $\text{p}\text{(n,m)=(}\text{1}\text{n}\text{∑}{}_{\mathrm{<mtext>d</mtext><mtext>m</mtext>}}\text{φ(d)(}{}_{\mathrm{<mtext>n</mtext><mtext>d</mtext>}}{}^{\mathrm{<mtext>m</mtext><mtext>d</mtext>}}\text{)}$, where φ is Euler's φ-function. $\text{P}\text{(n, m)}$ is shown to be set-isomorphic to the family of orbits $\text{C}\text{(n, m)}$ of the set of all m-subsets of an n-set acted upon by the cyclic permutation (12…n). This isomorphism yields an efficient method for determining the complete weight enumerator of any code generated by a circulant matrix.     

Polynomial stability of semigroups generated by operator matrices

In this paper, we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretical results are used to derive conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equation.     

Rings of matrices generated by a companion matrix

The regular representation of the quotient of a polynomial ring over the principal ideal determined by h(x) is the ring of matrices generated by the companion matrix of h(x). Properties of such rings, also called Barnett matrix rings, will be investigated.     

Subalgebras of matrix algebras generated by companion matrices

Let f,g∈Z[X] be monic polynomials of degree n and let C,D∈M_n(Z) be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra Z〈C,D〉 to be a sublattice of finite index in the full integral lattice M_n(Z), in which case we compute the exact value of this index in terms of the resultant of f and g. If R is a commutative ring with identity we determine when R〈C,D〉=M_n(R), in which case a presentation for M_n(R) in terms of C and D is given.     

Spaces of matrices of fixed rank

The aim of this paper is to obtain bounds for the dimensions of spaces of matrices of fixed rank and in some special cases determine the precise values.