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Some inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse

Guanghui Cheng*, Qin Tan and Zhuande Wang

Author Affiliations

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China

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Journal of Inequalities and Applications 2013, 2013:65  doi:10.1186/1029-242X-2013-65


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/65


Received:31 July 2012
Accepted:24 January 2013
Published:21 February 2013

© 2013 Cheng et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, some new inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse are given. These inequalities are sharper than the well-known results. A simple example is shown.

AMS Subject Classification: 15A18, 15A42.

Keywords:
Hadamard product; M-matrix; inverse M-matrix; strictly diagonally dominant matrix; eigenvalue

1 Introduction

A matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M1">View MathML</a> is called a nonnegative matrix if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M2">View MathML</a>. A matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3">View MathML</a> is called a nonsingular M-matrix [1] if there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M4">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M5">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M6">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M7">View MathML</a> is a spectral radius of the nonnegative matrix B, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M8">View MathML</a> is the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M9">View MathML</a> identity matrix. Denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M10">View MathML</a> the set of all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M9">View MathML</a> nonsingular M-matrices. The matrices in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M12">View MathML</a> are called inverse M-matrices. Let us denote

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M13">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M14">View MathML</a> denotes the spectrum of A. It is known that [2]

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M15">View MathML</a>

is a positive real eigenvalue of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M16">View MathML</a> and the corresponding eigenvector is nonnegative. Indeed

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M17">View MathML</a>

if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M18">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M19">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M4">View MathML</a>.

For any two <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M9">View MathML</a> matrices <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M22">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M23">View MathML</a>, the Hadamard product of A and B is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M24">View MathML</a>. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M25">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M26">View MathML</a> is also an M-matrix [3].

A matrix A is irreducible if there does not exist a permutation matrix P such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M27">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M28">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M29">View MathML</a> are square matrices.

For convenience, the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M30">View MathML</a> is denoted by N, where n (≥3) is any positive integer. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M1">View MathML</a> be a strictly diagonally dominant by row, denote

Recently, some lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and an inverse M-matrix have been proposed. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33">View MathML</a>, for example, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M34">View MathML</a> has been proven by Fiedler et al. in [4]. Subsequently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M35">View MathML</a> was given by Fiedler and Markham in [3], and they conjectured that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M36">View MathML</a>. Song [5], Yong [6] and Chen [7] have independently proven this conjecture. In [8], Li et al. improved the conjecture <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M37">View MathML</a> when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M38">View MathML</a> is a doubly stochastic matrix and gave the following result:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M39">View MathML</a>

In [9], Li et al. gave the following result:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M40">View MathML</a>

Furthermore, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M41">View MathML</a>, they have obtained

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M42">View MathML</a>

i.e., under this condition, the bound of [9] is better than the one of [8].

In this paper, our motives are to improve the lower bounds for the minimum eigenvalue <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M43">View MathML</a>. The main ideas are based on the ones of [8] and [9].

2 Some preliminaries and notations

In this section, we give some notations and lemmas which mainly focus on some inequalities for the entries of the inverse M-matrix and the strictly diagonally dominant matrix.

Lemma 2.1[6]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3">View MathML</a>be a strictly diagonally dominant matrix by row, i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M45">View MathML</a>

If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M46">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M47">View MathML</a>

Lemma 2.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3">View MathML</a>be a strictly diagonally dominantM-matrix by row. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M46">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M50">View MathML</a>

Proof Firstly, we consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M3">View MathML</a> is a strictly diagonally dominant M-matrix by row. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M52">View MathML</a>, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M53">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M54">View MathML</a>

Since A is strictly diagonally dominant, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M55">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M56">View MathML</a>. Therefore, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M57">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M58">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M59">View MathML</a>. Let us define one positive diagonal matrix

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M60">View MathML</a>

Similarly to the proofs of Theorem 2.1 and Theorem 2.4 in [8], we can prove that the matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M61">View MathML</a> is also a strictly diagonally dominant M-matrix by row for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M52">View MathML</a>. Furthermore, by Lemma 2.1, we can obtain the following result:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M63">View MathML</a>

i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M64">View MathML</a>

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M65">View MathML</a> to get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M66">View MathML</a>

This proof is completed. □

Lemma 2.3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M67">View MathML</a>be a strictly diagonally dominant matrix by row and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M69">View MathML</a>

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M70">View MathML</a>. Since A is an M-matrix, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M4">View MathML</a>. By <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M72">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M73">View MathML</a>

Hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M74">View MathML</a>

or equivalently,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M75">View MathML</a>

Furthermore, by Lemma 2.2, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M76">View MathML</a>

i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M77">View MathML</a>

Thus the proof is completed. □

Lemma 2.4[10]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M78">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M79">View MathML</a>be positive real numbers. Then all the eigenvalues ofAlie in the region

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M80">View MathML</a>

Lemma 2.5[11]

If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M38">View MathML</a>is a doubly stochastic matrix, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M82">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M83">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M84">View MathML</a>.

3 Main results

In this section, we give two new lower bounds for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M43">View MathML</a> which improve the ones in [8] and [9].

Lemma 3.1If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68">View MathML</a>is a doubly stochastic matrix, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M88">View MathML</a>

Proof This proof is similar to the ones of Lemma 3.2 in [8] and Theorem 3.2 in [9]. □

Theorem 3.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68">View MathML</a>be a doubly stochastic matrix. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M91">View MathML</a>

Proof Firstly, we assume that A is irreducible. By Lemma 2.5, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M92">View MathML</a>

Denote

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M93">View MathML</a>

Since A is an irreducible matrix, we know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M94">View MathML</a>. So, by Lemma 2.4, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M95">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M96">View MathML</a>

or equivalently,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M97">View MathML</a>

Secondly, if A is reducible, without loss of generality, we may assume that A has the following block upper triangular form:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M98">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M99">View MathML</a> is an irreducible diagonal block matrix, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M100">View MathML</a>. Obviously, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M101">View MathML</a>. Thus the reducible case is converted into the irreducible case. This proof is completed. □

Theorem 3.2If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M67">View MathML</a>is a strictly diagonally dominant by row, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M103">View MathML</a>

Proof Since A is strictly diagonally dominant by row, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M104">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M105">View MathML</a>

or equivalently,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M106">View MathML</a>

(1)

So, we can obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M107">View MathML</a>

(2)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M108">View MathML</a>

Therefore, it is easy to obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M109">View MathML</a>

Obviously, we have the desired result

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M110">View MathML</a>

This proof is completed. □

Theorem 3.3If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M67">View MathML</a>is strictly diagonally dominant by row, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M112">View MathML</a>

Proof Since A is strictly diagonally dominant by row, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M104">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M114">View MathML</a>

i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M115">View MathML</a>

(3)

So, we can obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M116">View MathML</a>

(4)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M117">View MathML</a>

Therefore, it is easy to obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M118">View MathML</a>

Obviously, we have the desired result

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M119">View MathML</a>

 □

Remark 3.1 According to inequalities (1) and (3), it is easy to know that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M120">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M121">View MathML</a>

That is to say, the result of Lemma 2.2 is sharper than the ones of Theorem 2.1 in [8] and Lemma 2.2 in [9]. Moreover, the results of Theorem 3.2 and Theorem 3.3 are sharper than the ones of Theorem 3.1 in [8] and Theorem 3.3 in [9], respectively.

Theorem 3.4If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M33">View MathML</a>is strictly diagonally dominant by row, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M123">View MathML</a>

Proof This proof is similar to the one of Theorem 3.5 in [8]. □

Remark 3.2 According to inequalities (2) and (4), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M124">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M125">View MathML</a>

That is to say, the bound of Theorem 3.4 is sharper than the ones of Theorem 3.5 in [8] and Theorem 3.4 in [9], respectively.

Remark 3.3 Using the above similar ideas, we can obtain similar inequalities of the strictly diagonally M-matrix by column.

4 Example

For convenience, we consider the M-matrix A is the same as the matrix of [8]. Define the M-matrix A as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M126">View MathML</a>

1. Estimate the upper bounds for entries of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M68">View MathML</a>. Firstly, by Lemma 2.2(2) in [9], we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M128">View MathML</a>

By Lemma 2.2, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M129">View MathML</a>

By Lemma 2.3 and Theorem 3.1 in [9], we get

By Lemma 2.3 and Lemma 3.1, we get

2. Lower bounds for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M43">View MathML</a>.

By Theorem 3.2 in [9], we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M133">View MathML</a>

By Theorem 3.1, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/65/mathml/M134">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Acknowledgements

This research is supported by National Natural Science Foundations of China (No. 11101069).

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