Two cores of a nonnegative matrix

We prove that the sequence of eigencones (i.e., cones of nonnegative eigenvectors) of positive powers A^k$A^k$ of a nonnegative square matrix A is periodic both in max algebra and in nonnegative linear algebra. Using an argument of Pullman, we also show that the Minkowski sum of the eigencones of powers of A is equal to the core of A defined as the intersection of nonnegative column spans of matrix powers, also in max algebra. Based on this, we describe the set of extremal rays of the core.     

On QB ∞-Rings

In this article, we introduce a new class of rings, the QB _∞-rings. We establish various properties of this concept. These show that, in several respects, QB _∞-rings behave like QB-rings. We prove that the notion of QB _∞-rings is a Morita invariant property of rings and every finite subdirect product of QB _∞-rings is a QB _∞-ring. We also exhibit examples to point out that the class of QB _∞-rings is much larger than the class of QB-rings.     

Cone-transitive matrix semigroups

Semigroups of matrices (over an ordered field) with non-negative entries are considered. A complete characterization is obtained for the semigroups which are minimal transitive on the positive (or non-negative) cone of the underlying vector space. Consequently, an explicit form for the semigroups sharply transitive on the cone is derived.     

Sign Regular Matrices of Order Two

A matrix is called sign regular of order k if every minor of order i has the same sign for each i = 1,2,<, k . If an m × n matrix is sign regular of order k for k = min { m,n } then it is called sign regular. This paper studies some properties of sign regular matrices of order two. Remarkable properties are proved when the row sums of these matrices form a monotone vector.     

Positive and real-positive solutions to the equation axa*=c in C*-algebras

We obtain new representations for the general positive and real-positive solutions of the equation axa*=c in a C*-algebra using the characterization of positivity based on a matrix representation of an element and the generalized Schur complement. Applications to the equation AXA*=C for operators between Hilbert spaces and for finite matrices are given.     

Eigenvalues of products of matrices

Let A and B be n?×?n matrices over an algebraically closed field F. The pair ( A,?B ) is said to be spectrally complete if, for every sequence c₁,…,c_n ∈F such that det (AB)=c₁ ,…,c_n , there exist matrices A′,B,′∈F,^n×n similar to A,?B, respectively, such that A′B′ has eigenvalues c₁,…,c_n . In this article, we describe the spectrally complete pairs. Assuming that A and B are nonsingular, the possible eigenvalues of A′B′ when A′ and B′ run over the sets of the matrices similar to A and B, respectively, were described in a previous article.     

Covariate Selection in High-Dimensional Generalized Linear Models With Measurement Error

In many problems involving generalized linear models, the covariates are subject to measurement error. When the number of covariates p exceeds the sample size n, regularized methods like the lasso or Dantzig selector are required. Several recent papers have studied methods which correct for measurement error in the lasso or Dantzig selector for linear models in the p > n setting. We study a correction for generalized linear models, based on Rosenbaum and Tsybakov’s matrix uncertainty selector. By not requiring an estimate of the measurement error covariance matrix, this generalized matrix uncertainty selector has a great practical advantage in problems involving high-dimensional data. We further derive an alternative method based on the lasso, and develop efficient algorithms for both methods. In our simulation studies of logistic and Poisson regression with measurement error, the proposed methods outperform the standard lasso and Dantzig selector with respect to covariate selection, by reducing the number of false positives considerably. We also consider classification of patients on the basis of gene expression data with noisy measurements. Supplementary materials for this article are available online.     

On Algebraic Limits and Topological Properties of Sets of Möbius groups in $ \bar {R}^n $

Let G be the algebraic limit of a sequence $ \{ G_m\} $ of r generator subgroups of $ M(\bar {R}^n) $ . We prove that (i) if $ G_m $ is elementary, then G is elementary; (ii) if $ G_m $ is not non-elementary and non-discrete and G satisfies the Condition A, then G is not non-elementary and non-discrete. Let $ E_l = \{ (f_1, f_2,\ldots ,f_l): f_1,f_2,\ldots ,f_l\in M(\bar {R}^2), \langle \,f_1,f_2,\ldots , \,f_l \rangle\ elementary\} $ and $ D_l = \{ (f_1,f_2,\ldots ,f_l): f_1,f_2,\ldots ,f_l\in M(\bar {R}^2), \langle \,f_1,f_2,\ldots ,f_l \rangle\ discrete\} $ . we obtain that the set $ E_l , D_l\cup E_l $ and $ D_l\cap E_l^c $ are closed in $ \underbrace {M(\bar {R}^2)\times \cdots \times M(\bar {R}^2)}\limits _{l} $ .