Algebra and Combinatorics

 

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Author(s)
Américo Lopes Bento; António Leal Duarte;

Title
A Fiedler's type characterization of band matrices

Abstract
Let $\mathbb{K}$ be a field and p an integer positive number. We denote by $\mathcal{B}_{n}^{p}(\mathbb{K})$ the set ofn-by-n symmetric band matrices of bandwidth 2p-1, i.e., if $A=[a_{ij}]\in\mathcal{B}_{n}^{p}(\mathbb{K})$ then $a_{ij}=0$ if $|i-j|>p-1$. Let $\widehat{\mathcal{B}}_{n}^{p}(\mathbb{K})$ be the set of matrices from $\mathcal{B}_{n}^{p}(\mathbb{K})$ in which the entries $(i,j)$, |i-j|=p-1, are different from zero.

Let A be a n-by-n symmetric matrix with entries from $\mathbb{K}$; and p such that $3\leqslant p\leqslant n$. We will show that: $\mathrm{rank}(A+B)\geqslant n-p+1$, for every $B\in\mathcal{B}_{n}^{p-1}( \mathbb{K}) $, if and only if $A\in\widehat{\mathcal{B}}_{n}^{p}(\mathbb{K}) $.

Preprint series
Pré-publicações do Departamento de Matemática da Universidade de Coimbra

Issue
04-34

Year
2004

 
     
 

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