周 平，黃衛(wèi)華

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-矩陣逆矩陣的無窮大范數(shù)上界的進(jìn)一步研究

周 平，黃衛(wèi)華

(文山學(xué)院數(shù)學(xué)學(xué)院，云南文山663000)

根據(jù)-矩陣的性質(zhì)和無窮大范數(shù)的定義，得到嚴(yán)格對角占優(yōu)-矩陣逆矩陣的無窮大范數(shù)上界的估計(jì)式，并給出-矩陣的最小特征值下界的新估計(jì)式. 理論分析和算例表明，文章給出的兩個估計(jì)式改進(jìn)了現(xiàn)有文獻(xiàn)的估計(jì)算法.

-矩陣；對角占優(yōu)；無窮大范數(shù)；最小特征值

1 基本定義和引理

定義2[1-7]若可表示為. 其中是單位矩陣，，是非負(fù)實(shí)數(shù)且，則稱為-矩陣. 特別地，當(dāng)時，稱為奇異矩陣；當(dāng)時，稱為非奇異矩陣. 記所有階非奇異矩陣所組成的集合為.

定義3[1]設(shè)的特征值為，令，則叫做的譜；中模最大的，即稱為的譜半徑.

定義4[3]設(shè)，記，稱為的最小特征值且.

定義5[4-7]設(shè)，且滿足條件：1)，；2)； 3)，. 存在非零元素序列，其中，則稱為弱鏈對角占優(yōu)矩陣.

定義6[5]設(shè)，任取，有，則稱為-矩陣. 設(shè)，非空指標(biāo)集合，為行數(shù)和列數(shù)都是的的子矩陣. 令，其中. 例如表示刪去的第一行第一列得到的矩陣.

定義7[5-6]設(shè)，若，則稱為嚴(yán)格對角占優(yōu)矩陣.

引理1[5]設(shè)是′階弱鏈對角占優(yōu)-矩陣，則也是弱鏈對角占優(yōu)-矩陣，且存在，，.

引理3[7]設(shè)是嚴(yán)格對角占優(yōu)-矩陣，，則.

引理4[4]設(shè)是′階弱鏈對角占優(yōu)-矩陣，，，，則，，，. 其中.

(2)

由式(1)和(2)，得

.

.

，.

同理，根據(jù)定義4可得到下面的推論2.

注：由此推論可知，本文定理2改進(jìn)了文獻(xiàn)[6]中定理2的估計(jì)式.

.

3 算例

[1] 陳景良, 陳向暉. 特殊矩陣[M]. 北京: 清華大學(xué)出版社, 2001: 23-87.

[2] VARGA R S. On diagonal dominance arguments for bounding ||-1||￥[J]. Linear Algebra and its applications, 1976, 14(3): 211-217.

[3] CHENG G H, HUANG T Z. An upper bound for ||-1||￥of strictly diagonally dominant M-matrices[J]. Linear Algebra and its applications, 2007, 426(2-3): 667-673.

[4] SHIVAKUMAR P N, WILLIAMS J J, YE Q, et al. On two-sided bounds related to weakly diagonally dominant M-matrices with application to digital circuit dynamics[J]. SIAM Journal on Matrix Analysis and Applications, 1996, 17(2): 298-312.

[5] LI W. The infinity norm bound for the inverse of nonsingular diagonal dominant matrices[J]. Applications Math. Letter, 2007, 21(2): 258-263.

[6] HUANG T Z, ZHU Y. Estimation of ||-1||￥for weakly chained diagonally dominant M-matrices[J]. Linear Algebra and its applications, 2010, 432(2-3): 670-677.

[7] CHENG GUANGHUI, TAN QIN, WANG ZHUANDE. Some inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse[J]. Journal of Inequalities and Applications, 2013, 65(1): 1029-1035.

(責(zé)任編輯：饒 超)

Further Research on the Upper Bounds for the Infinity Norms of-matrices

ZHOU Ping, HUANG Weihua

(School of Mathematics, Wenshan University, Wenshan 663000, China)

According to the properties of-matrix and the definition of infinity norm, some upper bounds for strictly diagonally dominant-matrices are further researched, and the corresponding new results are given. At the same time new lower bounds on the smallest eigenvalue of-matrixis derived. Theory analysis and numerical figure showed that the theorem one and two in this paper improve the existing results in some cases.

-matrix; Diagonal dominance; Infinity norm; Minimum eigenvalue

O151.21

A

2095-4476(2015)05-0009-03

2014-12-15 ;

2015-01-08

云南省科技廳應(yīng)用基礎(chǔ)研究青年項(xiàng)目(2013FD052); 云南省教育廳項(xiàng)目(2013Y585); 文山學(xué)院重點(diǎn)學(xué)科數(shù)學(xué)建設(shè)項(xiàng)目(12WSXK01)

周 平(1987— ), 女, 云南永平人, 文山學(xué)院數(shù)學(xué)學(xué)院講師.