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# New bounds for the minimum eigenvalue of M-matrices

Feng Wang
• Corresponding author
• College of Science, Guizhou Minzu University, Guiyang, Guizhou 550025, China
• Email:
/ Deshu Sun
• College of Science, Guizhou Minzu University, Guiyang, Guizhou 550025, China
• Email:
Published Online: 2016-12-17 | DOI: https://doi.org/10.1515/math-2016-0091

## Abstract

Some new bounds for the minimum eigenvalue of M-matrices are obtained. These inequalities improve existing results, and the estimating formulas are easier to calculate since they only depend on the entries of matrices. Finally, some examples are also given to show that the bounds are better than some previous results.

MSC 2010: 15A18; 15A42

## 1 Introduction

Let Cn×n (Rn×n) denote the set of all n × n complex (real) matrices, A = (aij) ∈ Cn×n, N = {1,2,... ,n}. We write A ≥ 0 if all aij ≥ 0(i, jN). A is called nonnegative if A ≥ 0. Let Zn denote the class of all n × n real matrices all of whose off-diagonal entries are nonpositive. A matrix A is called an M-matrix [1] if A Zn and the inverse of A, denoted by A-1, is nonnegative. Mn will be used to denote the set of all n × n M-matrices.

Let A be an M-matrix. Then there exists a positive eigenvalue of A, τ (A) = ρ(A-1)-1, where ρ(A-1) is the spectral radius of the nonnegative matrix A-1, τ (A) = min{ |λ| : λ σ(A)}, σ(A) denotes the spectrum of A. τ (A) is called the minimum eigenvalue of A [2, 3].

The Hadamard product of two matrices A = (aij) ∈ Rn×n and B = (bij) ∈ Rn×n is the matrix αB = (aij bij) Rn×n.

An n × n matrix A is said to be reducible if there exists a permutation matrix P such that $PTAP=A110A21A22,$ where A11, A22 are square matrices of order at least one. We call A irreducible if it is not reducible. Note that any nonzero 1 × 1 matrix is irreducible.

A matrix A = (aij) ∈ Cn×n is called a weakly chained diagonally dominant M-matrix [4] if A satisfies the following conditions:

(i) For all i, j N with i j, aij ≤ 0 and aii > 0;

(ii) For all $i\in N,\left|{a}_{ii}\right|\ge \sum _{j=1,\ne i}^{n}\left|{a}_{ij}\right|\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}J\left(A\right)=\left\{i\in N:\left|{a}_{ii}\right|>\sum _{j=1,\ne i}^{n}\left|{a}_{ij}\right|\right\}\ne \mathrm{\varnothing };$

(iii) For all iN, i J (A), there exist indices i1, i2, ... ,ik in N with ${a}_{{i}_{r},}{i}_{r+1}\ne 0,\phantom{\rule{thinmathspace}{0ex}}0\le r\le k-1,$ where i0 = i and ik J (A).

Estimating the bounds for the minimum eigenvalue τ (A) of an M-matrix A is an interesting subject in matrix theory, and has important applications in many practical problems [4-6]. Hence, it is necessary to estimate the bounds for τ(α).

In [4], Shivakumar et al. obtained the following bound for τ (A): Let A = (aij) ∈ Rn×n be a weakly chained diagonally dominant M-matrix, and let A-1 = (αij). Then $mini∈N∑j∈Naij≤τ(A)≤mini∈N∑j∈Naij,τ(A)≤mini∈Naii,1M≤τ(A)≤1m,$(1) where $M=mini∈N∑j∈Nαij,m=mini∈N∑j∈Nαij.$

Subsequently, Tian et al. [7] provided a lower bound for τ(A) by using the spectral radius of the Jacobi iterative matrix Ja of A: Let A = (aij) ∈ Rn×n be an M-matrix and A-1 = (αij)· Then $τ(A)≥11+(n−1)ρ(JA)1mini∈N⁡{αii}.$(2)

Recently, Li et al. [8] improved (2) and gave the following result: Let B = (bij) Rn×n be an M-matrix and B-1 = (βij). Then $τ(B)≥2mini≠j⁡{βii+βjj+[(βii−βjj)2+4(n−1)2βiiβjjρ2(JB)]12}.$(3)

In this paper, we continue to research the problems mentioned above. For M-matrix B, we establish some new inequalities on the bounds for τ(B). Finally, some examples are given to illustrate our results.

For convenience, we employ the following notation throughout. Let A = (a¡j) be an η × η matrix. For i, j, k N, i j, denote $Ri=∑j≠i|aji|,di=Ri|aii|,sji=|aji|+∑k≠j,i|ajk|dk|ajj|;uji=aji+∑k≠j,i|ajk|Ski|ajj|,vji=aji+∑k≠j,i|ajk|uki|ajj|;hi=maxj≠iajiajjvij−∑k≠j,i|ajk|vki,wij=aji+∑k≠j,i|ajk|vkihiajj,wi=maxj≠iwij.$

## 2 Main results

In this section, we present our main results. Firstly, we give some notation and lemmas.

Let A ≥ 0 and D = diag(aij). Denote $C=A-D,{\mathcal{J}}_{A}={D}_{1}^{-1}C,{D}_{1}=diag\left({d}_{ii}\right),$ where $dii=1,ifaii=0,aii,ifaii≠0.$

By the definition of 𝒥A, we obtain $ρ(JAT)=ρ(D1−1CT)=ρ(CD1−1)=ρ(D1−1(CD1−1)D1)=ρ(D1−1C)=ρ(JA).$

Lemma 2.1: ([9]). Let A Cn×n, and let x1, x2,..., xn be positive real numbers. Then all the eigenvalues of A lie in the region $⋃i{z∈C:|z−aii|≤xi∑j≠i1xj|aji|,i∈N}.$

Lemma 2.2: ([3]). Let ACn×n, and let x1, x2, ... , xn be positive real numbers. Then all the eigenvalues of A lie in the region $⋃j≠j{z∈C:|z−aii||z−ajj|≤(xi∑κ≠i1xk|aki|)(xj∑l≠j1xl|alj|)}.$

Lemma 2.3: ([3]). Let A, BRn×n , and let X, YRn×n be diagonal matrices. Then $X(A∘B)Y=(XAY)∘B=(XA)∘(BY)=(AY)∘(XB)=A∘(XBY).$

Lemma 2.4: ([3]). Let A = (aij) ∈ Mn. Then there exists a positive diagonal matrix X such that X-1 AX is a strictly diagonally dominant M-matrix.

Lemma 2.5: Let A = (aij) ∈ Rn×n be a strictly diagonally dominant matrix and let A-1 = (αij). Then $αij≤wjiαjj≤wjαjj,j,i∈N,j≠i.$

Proof: This proof is similar to the one of Lemma 2.2 in [10].

Theorem 2.6: Let A = (aij) ≥ 0, B = (bij) ∈ Mn and let B-1 = (ßij). Then $ρ(A∘B−1)≤max1≤i≤n{(aii+wiρ(JA)dii)βii}.$(4)

Proof: It is evident that the result holds with equality for n = 1.We next assume that n ≥ 2.(1) First, we assume that A and B are irreducible matrices. Since B is an M-matrix, by Lemma 2.4, there exists a positive diagonal matrix X, such that X-1 BX is a strictly row diagonally dominant M-matrix, and $ρ(A∘B−1)=ρ(X−1(A∘B−1)X)=ρ(A∘(X−1BX)−1).$Hence, for convenience and without loss of generality, we assume that B is a strictly diagonally dominant matrix.On the other hand, since A is irreducible and so is ${\mathcal{J}}_{{A}^{T}}.$ Then there exists a positive vector x = (xi) such that ${\mathcal{J}}_{{A}^{T}}x=\rho \left({\mathcal{J}}_{{A}^{T}}\right)x=\rho \left({\mathcal{J}}_{A}\right)x,$ thus, we obtain $\sum _{j\ne i}{a}_{ji}{x}_{j}=\rho \left({\mathcal{J}}_{A}\right){d}_{ii}{x}_{i}.$ Let $\stackrel{~}{A}=\left({\stackrel{~}{a}}_{ij}\right)=XA{X}^{-1}$ in which X is the positive matrix X = diag(x1, x2,..., xn). Then, we have $A~=(a~ij)=XAX−1=a11a12x1x2⋯a1nx1a21x2x1a22⋯a2nx2xn⋮⋮⋱⋮an1xnx1an2xnx2⋯ann.$From Lemma 2.3, we have $A~∘B−1=(XAX−1)∘B−1=X(A∘B−1)X−1.$Thus, we obtain $\rho \left(\stackrel{~}{A}\circ {B}^{-1}\right)=\rho \left(A\circ {B}^{-1}\right).\phantom{\rule{thinmathspace}{0ex}}\text{Let}\phantom{\rule{thinmathspace}{0ex}}\lambda =\rho \left(\stackrel{~}{A}\circ {B}^{-1}\right),\phantom{\rule{thinmathspace}{0ex}}\text{so that}\phantom{\rule{thinmathspace}{0ex}}\lambda \ge {a}_{ii}{\beta }_{ii},\mathrm{\forall }i\in N.$ By Lemma 2.1, there exists i0 N, such that $|λ−ai0i0βi0i0|≤wi0∑t≠i01wta~ti0βti0≤wi0∑t≠i01wta~ti0wti0βi0i0$ $≤wi0∑t≠i0a~ti0βi0i0=wi0βi0i0∑t≠i0ati0xtxi0=wi0ρ(JA)di0i0βi0i0.$Therefore, $λ≤ai0i0βi0i0+wi0ρ(JA)di0i0βi0i0=(ai0i0+wi0ρ(JA)di0i0)βi0i0,$ i.e., $ρ(A∘B−1)≤(ai0i0+wi0ρ(JA)di0i0)βi0i0≤max1≤i≤n{(aii+wiρ(JA)dii)βii}.$(2) Now, assume that one of A and B is reducible. It is well known that a matrix in Zn is a nonsingular M-matrix if and only if all its leading principal minors are positive (see [1]). If we denote by τ = (tij) the n × n monomial matrix with t12 = t23 = · · · = tn-1, n = tn1 = -1, the remaining tij zero, then both A — ɛτ and B + ɛτ are irreducible matrices for any chosen positive real number ɛ, sufficiently small such that all the leading principal minors of B + ɛτ are positive. Now, we substitute A — ɛτ and B + ɛτ for A and B, respectively, in the previous case, and then letting ɛ → 0, the result follows by continuity.

Theorem 2.7: Let B = (bij) ∈ Mn and B-1 = (ßij). Then $τ(B)≥1max1≤i≤n{(1+wi(n−1))βii}.$(5)Proof. Let all entries of A in (4) be 1. Then aii = 1(∀iN), ρ(JA) = n - 1. Therefore, by (4), we have $τ(B)=1ρ(B−1)≥1max1≤i≤n{(1+wi(n−1))βii}.$The proof is completed.

Theorem 2.8: Let A = (aij) 0, B = (bij) Mn and let B-1 = (ßij). Then $ρ(A∘B−1)≤12maxi≠jaiiβii+ajjβjj+Δij,$(6) where ${\mathrm{\Delta }}_{ij}=\left[\left({a}_{ii}{\beta }_{ii}-{a}_{jj}{\beta }_{jj}{\right)}^{2}+4{w}_{i}{w}_{j}{\rho }^{2}\left({\mathcal{J}}_{A}\right){d}_{ii}{d}_{jj}{\beta }_{ii}{\beta }_{jj}{\right]}^{\frac{1}{2}}.$

Proof: It is evident that the result holds with equality for n = 1.We next assume that n ≥ 2. For convenience and without loss of generality, we assume that B is a strictly row diagonally dominant matrix.(i) First, we assume that A and B are irreducible matrices. Since A is irreducible and so is ${\mathcal{J}}_{{A}^{T}}.$ Then there exists a positive vector y = (y¡) such that ${\mathcal{J}}_{{A}^{T}}y=\rho \left({\mathcal{J}}_{{A}^{T}}\right)y=\rho \left({\mathcal{J}}_{A}\right)y,$ thus, we obtain $∑κ≠iakiyk=ρ(JA)diiyi,∑κ≠jakjyk=ρ(JA)djjYj.$Let $\stackrel{^}{A}=\left({\stackrel{^}{a}}_{ij}\right)=YA{Y}^{-1}$ in which Y is the positive matrix Y = diag(y1, y2,..., yn). Then, we have $A^=(a^ij)=YAY−1=a11a12y1y2⋯a1ny1yna21y2y1a22⋯a2ny2yn⋮⋮⋱⋮an1yny1an2yny2⋯ann.$From Lemma 2.3, we get $A^∘B−1=(YAY−1)∘B−1=Y(A∘B−1)Y−1.$Thus, we obtain $\rho \left(\stackrel{^}{A}\circ {B}^{-1}\right)=\rho \left(A\circ {B}^{-1}\right).$ Let $\lambda =\rho \left(\stackrel{^}{A}\mathrm{o}{B}^{-1}\right)$ so that λaii ßii (∀i N). By Lemma 2.2, there exist i0, j0 N, i0 j0 such that $|λ−ai0i0βi0i0||λ−aj0j0βj0j0|<_(wi0∑κ≠i01wka^ki0βki0)(wj0∑κ≠j01wka^kj0βkj0).$Note that $wi0∑κ≠i01wka^ki0βki0<_wi0∑κ≠i01wka^ki0wki0βi0i0<_wi0βi0i0∑κ≠i0a^ki0=wi0βi0i0ρ(JA)di0i0;wj0∑κ≠j01wka^kj0βkj0<_wj0∑κ≠j01wka^kj0wkj0βj0j0<_wj0βj0j0∑κ≠j0a^kj0=wj0βj0j0ρ(JA)dj0j0.$Hence, we obtain $λ≤12(ai0i0βi0i0+aj0j0βj0j0+Δj0j0),$i.e., $ρ(AoB−1)≤12(ai0i0βi0i0+aj0j0βj0j0+Δi0i0)≤12maxi≠j{aiiβii+ajjβjj+Δij},$where $\mathrm{\Delta }=\left[\left({a}_{ii}{\beta }_{ii}-{a}_{jj}{\beta }_{jj}{\right)}^{2}+4{w}_{i}{w}_{j}{\rho }^{2}\left({\mathcal{J}}_{A}\right)\left({d}_{ii}{d}_{jj}{\beta }_{ii}{\beta }_{jj}{\right]}^{\frac{1}{2}}.$(ii) Now, assume that one of A and B is reducible. We substitute A — ɛτ and B + ɛτ for A and B, respectively, in the previous case (as in the proof of Theorem 2.6), and then letting ɛ → 0, the result follows by continuity.

Theorem 2.9: Let B = (bij) ∈ Mn and B-1 = (βij). Then $τ(B)≥2maxi≠j⁡{βii+βjj+βjj},$(7)where ${\mathrm{\Delta }}_{ij}=\left[\left({\beta }_{ii}-{\beta }_{jj}{\right)}^{2}+4\left(n-1{\right)}^{2}{w}_{i}{w}_{j}{\beta }_{ii}{\beta }_{jj}{\right]}^{\frac{1}{2}}.$

Proof: Proof. Let all entries of A in (6) be 1. Then $aii=1(∀i∈N),ρ(JA)=n−1,Δij=[(βii−βjj)2+4(n−1)2wiwjβiiβjj]12.$Therefore, by (6), we have $τ(B)=1ρ(B−1)≥2maxi≠jβii+βjj+Δij.$The proof is completed.

Remark 2.10: We next give a simple comparison between (4) and (6), (5) and (7), respectively. For convenience and without loss of generality, we assume that for i, jN, i j, $ajjβjj+wjdjjβjjρ(JA)≤aiiβii+widiiβiiρ(JA),$ i.e. $wjdjjβjjρ(JA)≤aiiβii−ajjβjj+widiiβiiρ(JA).$Hence, $Δij=[(aiiβii−ajjβjj)2+4wiwjρ2(JA)diidjjβiiβjj]12$ $≤[(aiiβii−ajjβjj)2+4wiρ(JA)diiβii(aiiβii−ajjβjj+widiiβiiρ(JA))]12=aiiβii−ajjβjj+2widiiβiiρ(JA).$Further, we obtain $aiiβii+ajjβjj+Δij≤2aiiβii+2widiiβiiρ(JA),$ by (6), $ρ(A∘B−1)≤12maxi≠j{ajjβii+ajjβjj+Δij}≤max1≤i≤n{(aii+wiρ(JA)dii)βii},$So, the bound in (6) is better than the bound in (4). Similarly, we can prove that the bound in (7) is better than the bound in (5).

## 3 Numerical examples

In this section, we present numerical examples to illustrate the advantages of our derived results.

Example 3.1: Let $B=1.1−0.4−0.4−0.31−0.5−0.3−0.20.6.$It is easy to see that B is an M-matrix. By calculations with Matlab 7.1, we haveτ(B) ≥ 0.10000000 (by (1)), τ(B) ≥ 0.09701726 (by (2)), τ(B) ≥ 0.12770275 (by (3)), τ(B) ≥ 0.10389610 (by (5)), τ(B) ≥ 0.13733592 (by (7)),respectively. In fact, τ(B) = 0.15010895. It is obvious that the bound in (7) is the best result.

Example 3.2: Let $B=1−0.2−0.1−0.3−0.1−0.41−0.2−0.1−0.1−0.3−0.21.2−0.1−0.2−0.2−0.3−0.31−0.1−0.1−0.3−0.1−0.21.$It is easy to see that B is an M-matrix. By calculations with Matlab 7.1, we haveτ(B) ≥ 0.10000000 (by (1)), τ(B) ≥ 0.15469345 (by (2)), τ(B) ≥ 0.15975146 (by (3)), τ(B) ≥ 0.18290441 (by (5)), τ(B) ≥ 0.24600529 (by (7)),respectively. In fact, τ(B) = 0.25174938. It is obvious that the bound in (7) is the best result.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China (11501141, 11361074), the Foundation of Science and Technology Department of Guizhou Province ([2015]2073,[2015]7206), and the Natural Science Programs of Education Department of Guizhou Province ([2015]420).

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Accepted: 2016-09-24

Published Online: 2016-12-17

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, Export Citation