一個含有2n個非零元的極小譜任意符號模式矩陣

趙麗娟，邵燕靈

(中北大學(xué) 理學(xué)院，山西 太原 030051)

一個含有2n個非零元的極小譜任意符號模式矩陣

趙麗娟，邵燕靈

(中北大學(xué) 理學(xué)院，山西 太原 030051)

研究了一個含有2n個非零元的符號模式矩陣，并運用冪零—雅可比方法和冪零—中心化方法證明該符號模式是極小譜任意的.

符號模式；譜任意；冪零—雅可比；冪零—中心化

0 引 言

引理 2[4](冪零-中心化方法)設(shè)S是n×n符號模式，B是S的一個指數(shù)為n的冪零實現(xiàn).如果B的中心中滿足條件C°BT=0的矩陣C只能是零矩陣，那么，S及其每一個母模式都是譜任意的.

1 主要結(jié)果

定理1當(dāng)n≥7時，S的所有母模式都是譜任意的.

其中ai<0,i=1,...,n-4,n，aj>0,j=n-3,n-2,n-1.下面分別用兩種不同的方法證明S的所有母模式都是譜任意的.

將上式第i行的λ倍加到第i+1行，i=1,2,...,n-1，然后再按第2,3,5,...,n-4,n-2,n-1,

n列依次展開，得：

(1)

所以

(2)

且

定理2S是極小譜任意的.

綜上所述，S是極小譜任意符號模式.

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[責(zé)任編輯：王軍]

A class of minimally spectrally arbitrary pattern matrix with 2n nonzero entries

ZHAO Lijuan, SHAO Yanling

(School of Science, North University of China, Taiyuan 030051, China)

In this paper we give a new minimally spectrally arbitrary patterns with2n nonzero entries.The sign pattern has been proved to be minimally spectrally arbitrary by using Nilpotent-Jacobian method and Nilpotent-Centralizer method.

sign pattern;spectrally arbitrary; nilpotent-Jacobian; nilpotent-centralizer

2014-12-09

山西省回國留學(xué)人員科研資助項目(12-070)

趙麗娟(1989-),女,山西大同人,中北大學(xué)碩士研究生,主要從事組合數(shù)學(xué)方面的研究.

O157

A

1672-3600(2015)09-0007-04