﻿ 随机矩阵非1特征值的新包含区域 The New Inclusion Region of Eigenvalue Different from 1 for a Stochastic Matrix

Pure Mathematics
Vol.06 No.04(2016), Article ID:18153,7 pages
10.12677/PM.2016.64051

The New Inclusion Region of Eigenvalue Different from 1 for a Stochastic Matrix

Baoxing Zhou1, Huifang Wei2, Yaotang Li1*

1School of Mathematics and Statistics, Yunnan University, Kunming Yunnan

2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming Yunnan

Received: Jul. 12th, 2016; accepted: Jul. 26th, 2016; published: Jul. 29th, 2016

ABSTRACT

Two new inclusion regions of eigenvalue different from 1 of stochastic matrices are given by using the a-eigenvalue inclusion theorem and the theory of modified matrices; and two new sufficient conditions of stochastic matrices nonsingular are obtained. Numerical examples are given to show that the existing results are improved in some cases.

Keywords:Stochastic Matrices, a1-Matrices, Eigenvalue Different from 1, a-Eigenvalue Inclusion Theorem

1云南大学，数学与统计学院，云南 昆明

2云南财经大学，统计与数学学院，云南 昆明

1. 引言

Shen等在文 [6] 中通过给出随机矩阵非奇异的三个充分条件，得到了随机矩阵非1实特征值的三个包含集。随后，Li等在文 [7] 中推广了Shen的结果，得到如下定理。

2. 随机矩阵新的非1特征值包含定理

,

, ,

, (2.1)

(2.2)

(2.3)

，由引理2.4得，由定理2.3知

,

, ,.

, (2.4)

(2.5)

(2.6)

,由引理2. 4得,再由定理2. 3知

,

3. 随机矩阵非奇异的两个新充分条件

, ,

,

4. 数值例子

.

.

.

Figure 1. The comparison of Φ(A), Γstol(A), Θsto(A)

K=10; A=rand(k,k); A =inv(diag(sum(A')))*A.

Figure 2. The comparison of Φ(A), Bstol(A), Θsto(A)

Figure 3. The comparison of Γstol(A) and Φ(A)

Table 1. The comparison of Φ(A) and T(A)

Table 2. The comparison of T(A), Γstol(A) and Bstol(A)

K=10; A =rand(k,k); A =inv(diag(sum(A')))*A.

The New Inclusion Region of Eigenvalue Different from 1 for a Stochastic Matrix[J]. 理论数学, 2016, 06(04): 361-367. http://dx.doi.org/10.12677/PM.2016.64051

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