Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix

Jianxing Zhao; Caili Sang

doi:10.1515/math-2016-0008

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Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix

Jianxing Zhao and Caili Sang

Published/Copyright: February 13, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

Some convergent sequences of the lower bounds of the minimum eigenvalue for the Hadamard product of a nonsingular M-matrix B and the inverse of a nonsingular M-matrix A are given by using Brauer’s theorem. It is proved that these sequences are monotone increasing, and numerical examples are given to show that these sequences could reach the true value of the minimum eigenvalue in some cases. These results in this paper improve some known results.

Keywords: M-matrix; Hadamard product; Minimum eigenvalue; Lower bounds; Sequence

MSC 2010: 15A06; 15A18; 15A42

1 Introduction

For a positive integer n, N denotes the set ﹛1, 2, · · · , n﹜, and ℝ^n×n(ℂ^n×n) denotes the set of all n × n real (complex) matrices throughout.

A matrix A = [a_ij] Î ℝ^n×n is called a nonsingular M-matrix if a_ij ≤ 0, i, j Î N, i ≠ j, A is nonsingular and A⁻¹ ≥ 0 (see [1, p.133]). Denote by M_n the set of all n × n nonsingular M-matrices.

If A is a nonsingular M-matrix, then there exists a positive eigenvalue of A equal to τ(A) ≡ [ρ(A⁻¹)]⁻¹, where ρ(A⁻¹) is the perron eigenvalue of the nonnegative matrix A⁻¹. It is easy to prove that τ(A) = min﹛|ƛ| : ƛ Î σ(A)﹜, where σ(A) denotes the spectrum of A (see [2, p.357]).

A matrix A is called reducible if there exists a nonempty proper subset I ⊂ N such that a_ij = 0, ∀ i Î I, ∀ jI. If A is not reducible, then we call A irreducible (see [3, p.128]).

For two real matrices A = [a_ij] and B = [b_ij] of the same size, the Hadamard product of A and B is defined as the matrix A ⚬ B = [a_ijb_ij]. If A and B are two nonsingular M-matrices, then it was proved in [4, Proposition 3] that A ⚬ B⁻¹ is also a nonsingular M-matrix.

Let A = [a_ij] Î ℝ^n×n, a_ii ≠ 0. For i, j, k Î N, j ≠ i, denote

dj=∑k≠j|ajk||ajj|,sji=|aji|+∑k≠j,i|ajk|dk|ajj|,si=maxj≠i﹛sij﹜; uji=|aji|+∑k≠j,i|ajk|ski|ajj|,ui=maxj≠i﹛uij﹜.

Recently, some lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse have been proposed. Let A = [a_ij] Î M_n, it was proved in [5] that

0<τ(A∘A−1)≤1.

Subsequently, Fiedler and Markham [4] gave a lower bound on τ(A ⚬ A⁻¹),

τ(A∘A−1)≥1n,

and conjectured that

τ(A∘A−1)≥2n.

Chen [6], Song [7] and Yong [8] have independently proved this conjecture.

In 2007, Li et al. improved the conjecture τ(A∘A−1)≥2n when A⁻¹ is a doubly stochastic matrix and gave the following result (i.e., [9, Theorem 3.1]): Let A = [a_ij] Î M_n and A⁻¹ be a doubly stochastic matrix. Then

(1)τ(A∘A−1)≥mini∈N﹛aii−si∑j≠i|aij|1+∑j≠isji﹜.

In 2013, Zhou et al. [10] also obtained (1) under the same condition.

In 2015, Chen gave the following result (i.e., [11, Theorem 3.2]): Let A = [a_ij] Î M_n and A⁻¹ = [α_ij] be a doubly stochastic matrix. Then

(2)τ(A∘A−1)≥mini≠j12﹛aiiαii+ajjαjj −[(aiiαii−ajjαjj)2+4uiujαiiαjj(∑k≠i|aki|)(∑k≠j|akj|)]12﹜

and they have obtained

mini≠j12﹛aiiαii+ajjαjj−[(aiiαii−ajjαjj)2+4uiujαiiαjj(∑k≠i|aki|)(∑k≠j|akj|)]12﹜ ≥mini∈N﹛aii−si∑j≠i|aij|1+∑j≠isij﹜,

i.e., under this condition, the bound of (2) is better than the one of (1).

In this paper, we present some convergent sequences of the lower bounds of τ(B ⚬ A⁻¹) and τ(A ⚬ A⁻¹), which improve (2). Numerical examples show that these sequences could reach the true value of τ(A ⚬ A⁻¹) in some cases.

2 Some notations and lemmas

In this section, we first give the following notations, which will be useful in the following proofs.

Let A=[aij]∈ℝn×n,aii≠0. For i,j,k∈N,j≠i,t=1,2,…, denote

hi=maxj≠i﹛|aji||ajj|sji−∑k≠j,i|ajk|ski﹜,υji(0)=|aji|+∑k≠j,i|ajk|skihi|ajj|,pji(t)=|aji|+∑k≠j,i|ajk|υki(t−1)|ajj|,pi(t)=maxj≠i{pij(t)},hi(t)=maxj≠i﹛|aji||ajj|pji(t)−∑k≠j,i|ajk|pki(t)﹜,υji(t)=|aji|+∑k≠j,i|ajk|pki(t)hi(t)|ajj|.

Lemma 2.1.

If A = [a_ij] Î ℝ^n×nis strictly row diagonally dominant, then, for all i, j Î N, j ≠ i, t = 1, 2, … ,

(a) 1 > d_j ≥ s_ji ≥ u_ji ≥ υ_ji⁽⁰⁾ ≥ p_ji⁽¹⁾ ≥ υ_ji⁽¹⁾ ≥ p_ji⁽²⁾ ≥ υ_ji⁽²⁾ ≥ · · · ≥ p_ji^(t) ≥ υ_ji^(t) ≥ · · · ≥ 0;

(b) 1 ≥ h_i ≥ 0, 1 ≥ h_i^(t) ≥ 0.

Proof. Since A is a strictly row diagonally dominant matrix, that is, |ajj| < ∑k≠j|ajk|, j ∈ N,, obviously, 1 > d_j ≥ 0, j Î N. By the definitions of d_j, s_ji, we have 1 > d_j ≥ s_ji ≥ 0, j, i Î N, j ≠ i. And then, by the definitions of s_ji, u_ji, we have s_ji ≥ u_ji, j,i Î N, j ≠ i. Hence,

|aji||ajj|sji−∑k≠j,i|ajk|ski=|ajj|sji−∑k≠j,i|ajk|dk|ajj|sji−∑k≠j,i|ajk|ski≤1,

from the definition of h_i, we have 0 ≤ h_i ≤ 1, i Î N. Furthermore, by the definitions of u_ji, υ_ji⁽⁰⁾, we have u_ji ≥ υ_ji⁽⁰⁾, j, i Î N, j ≠ i.

Since hi=maxj≠i﹛|aji||ajj|sji−∑k≠j,i|ajk|ski﹜,i∈N,, we have

hi≥|aji||aji|sji−∑k≠j,i|ajk|ski, i.e., sjihi≥|aji|+∑k≠j,i|ajk|skihi|ajj|=υji(0),j,i∈N,j≠i.

From the definitions of υ_ji⁽⁰⁾, p_ji⁽¹⁾, we have υ_ji⁽⁰⁾ ≥ p_ji⁽¹⁾ ≥ 0, j, i Î N, j ≠ i.

Hence,

|aji||ajj|pji(1)−∑k≠j,i|ajk|pki(1)=|ajj|pji(1)−∑k≠j,i|ajk|υki(0)|ajj|pji(1)−∑k≠j,i|ajk|pki(1)≤1,

by the definition of h_i⁽¹⁾, we have 0 ≤ h_i⁽¹⁾ ≤ 1, i Î N.

Since hi(1)=maxj≠i﹛|aji||ajj|pji(1)−∑k≠j,i|ajk|pki(1)﹜,i∈N, we have

hi(1)≥|aji||ajj|pji(1)−∑k≠j,i|ajk|pki(1), i.e., pji(1)hi(1)≥|aji|+∑k≠j,i|ajk|pki(1)hi(1)|ajj|=υji(1),j,i∈N,j≠i.

By 0 ≤ h_i⁽¹⁾ ≤ 1, i Î N, we have p_ji⁽¹⁾ ≥ υ_ji⁽¹⁾ ≥ 0, j, i Î N, j ≠ i. From the definitions of υ_ji⁽¹⁾, p_ji⁽²⁾, we obtain υ_ji⁽¹⁾ ≥ p_ji⁽²⁾ ≥ 0, j, i Î N, j ≠ i.

In the same way as above, we can also prove that

pji(2)≥υji(2)≥⋯pji(t)≥υji(t)≥⋯≥0,1≥hi(t)≥0,j,i∈N,j≠i,t=2,3,….

The proof is completed. □

Using the same technique as the proof of Lemma 2.2 in [11], we can obtain the following lemma.

Lemma 2.2.

If A = [a_ji] Î M_nis a strictly row diagonally dominant matrix, then A⁻¹ = [α_ij] exists, and

αji≤|aji|+∑k≠j,i|ajk|υki(t)ajiαii=pji(t+1)αii,j,i∈N,j≠i,t=0,1,2,….

Lemma 2.3.

[12] Let A = [a_ij] Î ℂ^n×nand x₁, x₂, · · · , x_nbe positive real numbers. Then all the eigenvalues of A lie in the region

∪i,j=1i≠jn﹛z∈ℂ:|z−aii||z−ajj|(xi∑k≠i1xk|aki|)(xj∑k≠j1xk|akj|)﹜.

3 Main results

In this section, we give several convergent sequences for τ(B ⚬ A⁻¹) and τ(A ⚬ A⁻¹).

Theorem 3.1.

Let A = [a_ij], B = [b_ij] Î M_nand A⁻¹ = [α_ij]. Then, for t = 1, 2, …,

(3)τ(B∘A−1)≥mini≠j12﹛αiibii+αjjbjj −[(αiibii−αjjbjj)2+4pj(t)pj(t)αiiαjj(∑k≠i|bkj|)]12﹜=Ωt.

Proof. It is evident that the result holds with equality for n = 1.

We next assume that n ≥ 2. Since A Î M_n, there exists a positive diagonal matrix D such that D⁻¹AD is a strictly row diagonally dominant M-matrix, and

τ(B∘A−1)=τ(D−1(B∘A1)D)=τ(B∘(D−1AD)−1).

Therefore, for convenience and without loss of generality, we assume that A is a strictly row diagonally dominant matrix.

(a) First, we assume that A and B are irreducible matrices. Since A is irreducible, then 0 < p_i^(t) < 1, for any i Î N. Let τ(B ⚬ A⁻¹) = ƛ. Since ƛ is an eigenvalue of B ⚬ A⁻¹, then 0 < ƛ < b_iiα_ii. By Lemma 2.2 and Lemma 2.3, there is a pair (i, j) of positive integers with i ≠ j such that

(4)|λ−biiαii||λ−bjjαjj|≤(pi(t)∑k≠i1pk(t)|bkiαki|)(pj(t)∑k≠j1pk(t)|bkjαkj|) ≤(pi(t)∑k≠i1pk(t)|bkipki(t)αii|)(pj(t)∑k≠j1pk(t)|bkjpkj(t)αjj|) ≤(pi(t)∑k≠i1pk(t)|bkipk(t)αii|)(pj(t)∑k≠j1pk(t)|bkjpk(t)αjj|) =(pi(t)αii∑k≠i|bki|)(pj(t)αii∑k≠j|bkj|).

From inequality (4), we have

(5)(λ−biiαii)(λ−bjjαjj)≤pj(t)αiiαjj(∑k≠i|bki|)(∑k≠j|bkj|).

Thus, (5) is equivalent to

λ≥12﹛αiibii+αjjbjj−[(αiibii−αjjbjj)2+4pj(t)pj(t)αiiαjj(∑k≠i|bkj|)]12﹜,

that is

τ(B∘A−1)≥12﹛αiibii+αjjbjj−[(αiibii−αjjbjj)2+4pi(t)pj(t)αiiαjj(∑k≠j|bkj|)]12﹜ ≥mini≠j12﹛αiibii+αjjbjj−[(αiibii−αjjbjj)2+4pi(t)pj(t)αiiαjj(∑k≠i|bkj|)]12﹜.

(b) Now, assume that one of A and B is reducible. It is well known that a matrix in Z_n = ﹛A = [a_ij] Î ℝ^n×n : a_ij ≤ 0, i ≠ j﹜ is a nonsingular M-matrix if and only if all its leading principal minors are positive (see condition (E17) of Theorem 6.2.3 of [1]). If we denote by C = [c_ij] the n × n permutation matrix with c₁₂ = c₂₃ = · · · = c_n_–1,n = c_n₁ = 1, the remaining c_ij zero, then both A − ϵC and B − ϵC are irreducible nonsingular M-matrices for any chosen positive real number ϵ, sufficiently small such that all the leading principal minors of both A − ϵC and B − ϵC are positive. Now we substitute A − ϵC and B − ϵC for A and B, in the previous case, and then letting ϵ → 0, the result follows by continuity. □

Theorem 3.2.

The sequence ﹛Ω_t﹜, t = 1, 2, … obtained from Theorem 3.1 is monotone increasing with an upper bound τ(B ⚬ A⁻¹) and, consequently, is convergent.

Proof. By Lemma 2.1, we have p_ji^(t) ≥ p_ji^(t+1) ≥ 0, j, i Î N, j ≠ i, t = 1, 2, … , so by the definiton of p_i^(t), it is easy to see that the sequence ﹛p_i^(t)﹜ is monotone decreasing. Then ﹛Ω_t﹜ is a monotonically increasing sequence with an upper bound τ(B ⚬ A⁻¹). Hence, the sequence is convergent. □

If B = A, according to Theorem 3.1, the following corollary is established.

Corollary 3.3.

Let A = [a_ij] Î M_nand A⁻¹ = [α_ij]. Then, for t = 1, 2, …,

(6)τ(A∘A−1)≥mini≠j12﹛aiiαii−ajjαjj −[(aiiαii−ajjαjj)2+4pi(t)pj(t)αiiαjj(∑k≠j|akj|)]12﹜=ϒt.

Remark 3.4.

We give a simple comparison between (2) and (6). According to Lemma 2.1, we know that u_ji ≥ p_ji^(t), j, i Î N, j ≠ i, t = 1, 2, …. Furthermore, by the definitions of u_i, p_i^(t), we have u_i ≥ p_i^(t), i Î N, t = 1, 2, …. Obviously, for t = 1, 2, …, the bounds in (6) are bigger than the bound in (2).

Similar to the proof of Theorem 3.1, Theorem 3.2 and Corollary 3.3, we can obtain Theorem 3.5, Theorem 3.6 and Corollary 3.7, respectively.

Theorem 3.5.

Let A = [a_ij], B = [b_ij] Î M_nand A⁻¹ = [α_ij]. Then, for t = 1, 2, …,

τ(B∘A−1)≥mini≠j12﹛αiibii+αjjbjj −[(αiibii−αjjbjj)2+4sisjαiiαjj(∑k≠i|bki|pki(t)sk)(∑k≠j|bkj|pkj(t)sk)]12﹜=Δt.

Theorem 3.6.

The sequence ﹛Δ_t﹜, t = 1, 2, … obtained from Theorem 3.5 is monotone increasing with an upper bound τ(B ⚬ A⁻¹) and, consequently, is convergent.

Corollary 3.7.

Let A = [a_ij] Î M_nand A⁻¹ = [α_ij]. Then, for t = 1, 2, …,

τ(A∘A−1)≥mini≠j12﹛αiiaii+αjjajj −[(αiiaii−αjjajj)2+4sisjαiiαjj(∑k≠i|aki|pki(t)sk)(∑k≠j|akj|pkj(t)sk)]12﹜=Γt.

Let L_t = max﹛ _t, Γ_t﹜. By Corollary 3.3 and Corollary 3.7, the following theorem is easily found.

Theorem 3.8.

Let A = [a_ij] Î M_nand A⁻¹ = [α_ij]. Then, for t = 1, 2, …,

τ(A∘A−1)≥Lt.

4 Numerical examples

In this section, several numerical examples are given to verify the theoretical results.

Example 4.1.

Consider the following M-matrix:

A=[20−1−2−3−4−1−1−3−2−2−118−3−1−1−4−2−1−3−1−2−110−1−1−10−1−1−1−3−1016−4−2−1−1−1−2−1−30−215−1−1−1−2−3−3−2−1−1−112−20−10−1−3−1−10−190−10−3−1−1−4100120−1−2−4−1−1−10−1−3140−3−10−1−1−10−1−211].

Since Ae = e and A^Te = e, e = [1, 1, · · ·, 1]^T, A⁻¹is doubly stochastic. Numerical results are given in Table 1 for the total number of iterations T = 10. In fact, τ(A ⚬ A⁻¹) = 0.9678.

Table 1.

The lower upper of τ(A ⚬ A⁻¹)

Method	t	L_t
Corollary 2.5 of [13]		0.1401
Conjecture of [4]		0.2000
Theorem 3.1 of [9]		0.2519
Theorem 3.2 of [15]		0.4125
Theorem 3.1 of [14]		0.4471
Theorem 3.2 of [11]		0.4732
Corollary 3 of [16]		0.6064
Theorem 3.8	t = 1	0.7388
	t = 2	0.8553
	t = 3	0.9059
	t = 4	0.9261
	t = 5	0.9346
	t = 6	0.9383
	t = 7	0.9400
	t = 8	0.9407
	t = 9	0.9409
	t = 10	0.9409

Remark 4.2.

Numerical results in Table 1 show that:

(a) Lower bounds obtained from Theorem 3.8 are bigger than these corresponding bounds in [4,9,11,13-16].

(b) Sequence obtained from Theorem 3.8 is monotone increasing.

Example 4.3.

A nonsingular M-matrix A = [a_ij] Î ℝ^n×nwhose inverse is doubly stochastic, is randomly generated by Matlab 7.1 (with 0-1 average distribution).

The numerical results obtained for T = 500 are listed in Table 2, where T are defined in Example 4.1.

Remark 4.4.

Numerical results in Table 2 show that it is effective by Theorem 3.8 to estimate τ(A ⚬ A⁻¹) for large order matrices.

Table 2.

The lower upper of τ(A ⚬ A⁻¹)

Method	t	L_t(n = 200)	L_t(n = 500)
Theorem 3.8	t = 1	0.0319	0.0133
	t = 30	0.3953	0.1972
	t = 60	0.6065	0.3452
	t = 90	0.7293	0.4647
	t = 120	0.8016	0.5609
	t = 150	0.8428	0.6384
	t = 180	0.8647	0.7011
	t = 210	0.8773	0.7519
	t = 240	0.8844	0.7928
	t = 270	0.8885	0.8255
	t = 300	0.8909	0.8520
	t = 330	0.8923	0.8734
	t = 360	0.8930	0.8908
	t = 390	0.8935	0.9049
	t = 420	0.8937	0.9163
	t = 450	0.8938	0.9249
	t = 480	0.8939	0.9316
	t = 500	0.8940	0.9352

Example 4.5.

Let A = [a_ij] Î ℝ^n×n, where a₁₁ = a₂₂ = · · · = a_n,n = 2, a₁₂ = a₂₃ = · · · = a_n_–1,n = a_n,₁ = −1, and a_ij = 0 elsewhere.

It is easy to know that A is a nonsingular M-matrix. If we apply Theorem 3.8 for n = 10 and n = 100, we have τ(A ⚬ A⁻¹) ≥ 0.7507 and τ(A ⚬ A⁻¹) ≥ 0.7500 when t = 1, respectively. In fact, τ(A ⚬ A⁻¹) = 0.7507 for n = 10 and τ(A ⚬ A⁻¹) = 0.7500 for n = 100.

Remark 4.6.

Numerical results in Example 4.5 show that the lower bound obtained from Theorem 3.8 could reach the true value of τ(A ⚬ A⁻¹) in some cases.

5 Conclusions

In this paper, we present a convergent sequence ﹛L_t﹜, t = 1, 2, … to approximate τ(A ⚬ A⁻¹). Although we do not give the error analysis, i.e., how accurately these bounds can be computed, from the numerical experiments of Example 4.3, we are pleased to see that the bounds are still on the increase with the increase of the number of iterations. At present, it is very difficult for the authors to give the error analysis. Next, we will study this problem.

Acknowledgement

The authors are very indebted to the reviewers for their valuable comments and corrections, which improved the original manuscript of this paper. This work is supported by the National Natural Science Foundations of China (Nos.11361074,11501141), Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073), Scientific Research Foundation for the introduction of talents of Guizhou Minzu university (No.15XRY003) and Scientific Research Foundation of Guizhou Minzu university (No.15XJS009).

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Received: 2015-10-26

Accepted: 2016-2-2

Published Online: 2016-2-13

Published in Print: 2016-1-1

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