Reduction of matrices over simple Ore domains

An informal introduction to perturbations of matrices determined up to similarity or congruence

The reductions of a square complex matrix A to its canonical forms under transformations of similarity, congruence, or *congruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the entries of A. We survey results about their behavior under perturbations of A and about normal forms of all matrices A+E in a neighborhood of A with respect to similarity, congruence, or *congruence. These normal forms are called miniversal deformations of A; they are not uniquely determined by A+E, but they are simple and depend continuously on the entries of…

A gradient method for inconsistency reduction of pairwise comparisons matrices

We investigate an application of a mathematically robust minimization method -- the gradient method -- to the consistencization problem of a pairwise comparisons (PC) matrix. Our approach sheds new light on the notion of a priority vector and leads naturally to the definition of instant priority vectors. We describe a sample family of inconsistency indicators based on various ways of taking an average value, which extends the inconsistency indicator based on the "$\sup$"- norm. We apply this family of inconsistency indicators both for additive and multiplicative PC matrices to show that the ch…

Wedderburn polynomials over division rings, II

A polynomial $f(t)$ in an Ore extension $K[t;S,D]$ over a division ring $K$ is a Wedderburn polynomial if $f(t)$ is monic and is the minimal polynomial of an algebraic subset of $K$. These polynomials have been studied in "Wedderburn polynomials over division rings,I (Journal of Pure and Applied Algebra, Vol. 186, (2004), 43-76). In this paper, we continue this study and give some applications to triangulation, diagonalization and eigenvalues of matrices over a division ring in the general setting of $(S,D)$-pseudo-linear transformations. In the last section we introduce and study the notion o…

On stable range one matrices

For 2 by 2 matrices over commutative rings, we prove a characterization theorem for left stable range 1 elements, we show that the stable range 1 property is left-right symmetric (also) at element level, we show that all matrices with one zero row (or zero column) over Bezout rings have stable range 1. Using diagonal reduction, we characterize all the 2 by 2 integral matrices which have stable range 1 and discuss additional properties including Jacobson Lemma for stable range 1 elements. Finally, we give an example of exchange stable range 1 integral 2 by 2 matrix which is not clean.

在条目置换下具有整数特征值的方阵

Square matrices with integer eigenvalues under entry permutations

研究整数多重集在任意方阵排列下是否总能产生整数特征值的问题，结合模约化与二进准则及块嵌入、秩约化方法，给出了零条目情形的完全刻画及非零情形下的部分精确准则。