New bounds for the minimum eigenvalue of M-matrices

Feng Wang; Deshu Sun

doi:10.1515/math-2016-0091

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New bounds for the minimum eigenvalue of M-matrices

Feng Wang and Deshu Sun

Published/Copyright: December 17, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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.

Key words: M-matrix; Nonnegative matrix; Hadamard product; Spectral radius; Minimum eigenvalue

MSC 2010: 15A18; 15A42

1 Introduction

Let C^n×n (R^n×n) denote the set of all n × n complex (real) matrices, A = (a_ij) ∈ C^n×n, N = {1,2,... ,n}. We write A ≥ 0 if all a_ij ≥ 0(i, j ∈ N). A is called nonnegative if A ≥ 0. Let Z_n 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 ∈ Z_n and the inverse of A, denoted by A^-1, is nonnegative. M_n 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 = (a_ij) ∈ R^n×n and B = (b_ij) ∈ R^n×n is the matrix α ∘ B = (a_ij b_ij) ∈ R^n×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 = (a_ij) ∈ C^n×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, a_ij ≤ 0 and a_ii > 0;

(ii) For all i∈N,aii≥∑j=1,≠inaijandJA=i∈N:aii>∑j=1,≠inaij≠∅;

(iii) For all i ∈ N, i ∉ J (A), there exist indices i₁, i₂, ... ,i_k in N with air,ir+1≠0,0≤r≤k−1, where i₀ = i and i_k ∈ 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 = (a_ij) ∈ R^n×n be a weakly chained diagonally dominant M-matrix, and let A^-1 = (α_ij). 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 J_a of A: Let A = (a_ij) ∈ R^n×n be an M-matrix and A^-1 = (α_ij)· 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 = (b_ij) ∈R^n×n be an M-matrix and B^-1 = (β_ij). 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(a_ij). Denote C=A−D,JA=D1−1C,D1=diag(dii), where

dii=1,ifaii=0,aii,ifaii≠0.

By the definition of 𝒥_A, 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∈C^n×n, and let x₁, x₂,..., x_n 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 A ∈ C^n×n, and let x₁, x₂, ... , x_n 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, B ∈ R^n×n , and let X, Y ∈ R^n×n be diagonal matrices. Then

X(A∘B)Y=(XAY)∘B=(XA)∘(BY)=(AY)∘(XB)=A∘(XBY).

Lemma 2.4

([3]). Let A = (a_ij) ∈ M_n. Then there exists a positive diagonal matrix X such that X^-1AX is a strictly diagonally dominant M-matrix.

Lemma 2.5

Let A = (a_ij) ∈ R^n×n be a strictly diagonally dominant matrix and let A^-1 = (α_ij). 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 = (a_ij) ≥ 0, B = (b_ij) ∈ M_n and let B^-1 = (ß_ij). 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^-1BX 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 JAT. Then there exists a positive vector x = (x_i) such that JATx=ρ(JAT)x=ρ(JA)x, thus, we obtain ∑j≠iajixj=ρ(JA)diixi.

Let A~=(a~ij)=XAX−1 in which X is the positive matrix X = diag(x₁, x₂,..., x_n). 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 ρ(A~∘B−1)=ρ(A∘B−1).Letλ=ρ(A~∘B−1),so thatλ≥aiiβii,∀i∈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 Z_n is a nonsingular M-matrix if and only if all its leading principal minors are positive (see [1]). If we denote by τ = (t_ij) the n × n monomial matrix with t₁₂ = t₂₃ = · · · = t_n-1, n = t_n1 = -1, the remaining t_ij 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 = (b_ij) ∈ M_n and B^-1 = (ß_ij). Then

τ(B)≥1max1≤i≤n{(1+wi(n−1))βii}.(5)

Proof. Let all entries of A in (4) be 1. Then a_ii = 1(∀i ∈ N), ρ(J_A) = 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 = (a_ij) ≥ 0, B = (b_ij) ∈M_n and let B^-1 = (ß_ij). Then

ρ(A∘B−1)≤12maxi≠jaiiβii+ajjβjj+Δij,(6)

whereΔij=[(aiiβii−ajjβjj)2+4wiwjρ2(JA)diidjjβiiβjj]12.

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 JAT. Then there exists a positive vector y = (y¡) such that JATy=ρ(JAT)y=ρ(JA)y, thus, we obtain

∑κ≠iakiyk=ρ(JA)diiyi,∑κ≠jakjyk=ρ(JA)djjYj.

Let A^=(a^ij)=YAY−1 in which Y is the positive matrix Y = diag(y₁, y₂,..., y_n). 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 ρ(A^∘B−1)=ρ(A∘B−1). Let λ=ρ(A^oB−1) so that λ ≥ a_ii ß_ii (∀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 Δ=[(aiiβii−ajjβjj)2+4wiwjρ2(JA)(diidjjβiiβjj]12.

(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 = (b_ij) ∈ M_n and B^-1 = (β_ij). Then

τ(B)≥2maxi≠j⁡{βii+βjj+βjj},(7)

where Δij=[(βii−βjj)2+4(n−1)2wiwjβiiβjj]12.

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, j ∈ N, 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 have

τ(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 have

τ(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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Received: 2015-12-9

Accepted: 2016-9-24

Published Online: 2016-12-17

Published in Print: 2016-1-1

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