Open Access

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

Journal of Inequalities and Applications20132013:65

DOI: 10.1186/1029-242X-2013-65

Received: 31 July 2012

Accepted: 24 January 2013

Published: 21 February 2013

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 = ( a i j ) R n × n is called a nonnegative matrix if a i j 0 . A matrix A R n × n is called a nonsingular M-matrix [1] if there exist B 0 and s > 0 such that
A = s I n B and s > ρ ( B ) ,
where ρ ( B ) is a spectral radius of the nonnegative matrix B, I n is the n × n identity matrix. Denote by M n the set of all n × n nonsingular M-matrices. The matrices in M n 1 : = { A 1 : A M n } are called inverse M-matrices. Let us denote
τ ( A ) = min { Re λ : λ σ ( A ) } ,
and σ ( A ) denotes the spectrum of A. It is known that [2]
τ ( A ) = 1 ρ ( A 1 )
is a positive real eigenvalue of A M n and the corresponding eigenvector is nonnegative. Indeed
τ ( A ) = s ρ ( B ) ,

if A = s I n B , where s > ρ ( B ) , B 0 .

For any two n × n matrices A = ( a i j ) and B = ( b i j ) , the Hadamard product of A and B is A B = ( a i j b i j ) . If A , B M n , then A B 1 is also an M-matrix [3].

A matrix A is irreducible if there does not exist a permutation matrix P such that
P A P T = [ A 1 , 1 A 1 , 2 0 A 2 , 2 ] ,

where A 1 , 1 and A 2 , 2 are square matrices.

For convenience, the set { 1 , 2 , , n } is denoted by N, where n (≥3) is any positive integer. Let A = ( a i j ) R n × n be a strictly diagonally dominant by row, denote
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equf_HTML.gif
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 M n , for example, τ ( A A 1 ) 1 has been proven by Fiedler et al. in [4]. Subsequently, τ ( A A 1 ) > 1 n was given by Fiedler and Markham in [3], and they conjectured that τ ( A A 1 ) > 2 n . Song [5], Yong [6] and Chen [7] have independently proven this conjecture. In [8], Li et al. improved the conjecture τ ( A A 1 ) 2 n when A 1 is a doubly stochastic matrix and gave the following result:
τ ( A A 1 ) min i { a i i s i R i 1 + j i s j i } .
In [9], Li et al. gave the following result:
τ ( A A 1 ) min i { a i i m i R i 1 + j i m j i } .
Furthermore, if a 11 = a 22 = = a n n , they have obtained
min i { a i i m i R i 1 + j i m j i } min i { a i i s i R i 1 + j i s j i } ,

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 τ ( A A 1 ) . 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 R n × n be a strictly diagonally dominant matrix by row, i.e.,
| a i i | > j i | a i j | , i N .
If A 1 = ( b i j ) , then
| b j i | k j | a j k | | a j j | | b i i | , j i , j N .
Lemma 2.2 Let A R n × n be a strictly diagonally dominant M-matrix by row. If A 1 = ( b i j ) , then
b j i | a j i | + k j , i | a j k | m k i a j j b i i u j b i i , j i , i N .
Proof Firstly, we consider A R n × n is a strictly diagonally dominant M-matrix by row. For i N , let
r i ( ε ) = max j i { | a j i | + ε a j j k j , i | a j k | }
and
m j i ( ε ) = r i ( ε ) ( k j , i | a j k | + ε ) + | a j i | a j j , j i .
Since A is strictly diagonally dominant, then r j i < 1 and m j i < 1 . Therefore, there exists ε > 0 such that 0 < r i ( ε ) < 1 and 0 < m j i ( ε ) < 1 . Let us define one positive diagonal matrix
M i ( ε ) = diag ( m 1 i ( ε ) , , m i 1 , i ( ε ) , 1 , m i + 1 , i ( ε ) , , m n i ( ε ) ) .
Similarly to the proofs of Theorem 2.1 and Theorem 2.4 in [8], we can prove that the matrix A M i ( ε ) is also a strictly diagonally dominant M-matrix by row for any i N . Furthermore, by Lemma 2.1, we can obtain the following result:
m j i 1 ( ε ) b j i | a j i | + k j , i | a j k | m k i ( ε ) m j i ( ε ) a j j b i i , j i , j N ,
i.e.,
b j i | a j i | + k j , i | a j k | m k i ( ε ) a j j b i i , j i , j N .
Let ε 0 + to get
b j i | a j i | + k j , i | a j k | m k i a j j b i i u j b i i , j i , j N .

This proof is completed. □

Lemma 2.3 Let A = ( a i j ) M n be a strictly diagonally dominant matrix by row and A 1 = ( b i j ) , then we have
1 a i i b i i 1 a i i j i | a i j | u j i , i N .
Proof Let B = A 1 . Since A is an M-matrix, then B 0 . By A B = B A = I n , we have
1 = j = 1 n a i j b j i = a i i b i i j i | a i j | b j i , i N .
Hence
1 a i i b i i , i N ,
or equivalently,
1 a i i b i i , i N .
Furthermore, by Lemma 2.2, we get
1 = a i i b i i j i | a i j | b j i ( a i i j i | a i j | u j i ) b i i , i N ,
i.e.,
b i i 1 a i i j i | a i j | u j i , i N .

Thus the proof is completed. □

Lemma 2.4 [10]

Let A C n × n and x 1 , x 2 , , x n be positive real numbers. Then all the eigenvalues of A lie in the region
1 n { z C : | z a i i | x i j i 1 x j | a j i | } .

Lemma 2.5 [11]

If A 1 is a doubly stochastic matrix, then A e = e , A T e = e , where e = ( 1 , 1 , , 1 ) T .

3 Main results

In this section, we give two new lower bounds for τ ( A A 1 ) which improve the ones in [8] and [9].

Lemma 3.1 If A M n and A 1 = ( b i j ) is a doubly stochastic matrix, then
b i i 1 1 + j i u j i , i N .

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

Theorem 3.1 Let A M n and A 1 = ( b i j ) be a doubly stochastic matrix. Then
τ ( A A 1 ) min i { a i i u i R i 1 + j i u j i } .
Proof Firstly, we assume that A is irreducible. By Lemma 2.5, we have
a i i = j i | a i j | + 1 = j i | a j i | + 1 and a i i > 1 , i N .
Denote
u j = max i j { u j i } = max { | a j i | + k j , i | a j k | m k i a j j } , j N .
Since A is an irreducible matrix, we know that 0 < u j 1 . So, by Lemma 2.4, there exists i 0 N such that
| λ a i 0 i 0 b i 0 i 0 | u i 0 j i 0 1 u j | a j i 0 b j i 0 | ,
or equivalently,
| λ | a i 0 i 0 b i 0 i 0 u i 0 j i 0 1 u j | a j i 0 b j i 0 | a i 0 i 0 b i 0 i 0 u i 0 j i 0 1 u j | a j i 0 | u j b i 0 i 0 (by Lemma 2.2) ( a i 0 i 0 u i 0 j i 0 | a j i 0 | ) b i 0 i 0 = ( a i 0 i 0 u i 0 R i 0 ) b i 0 i 0 a i 0 i 0 u i 0 R i 0 1 + j i 0 u j i 0 (by Lemma 3.1) min i { a i i u i R i 1 + j i u j i } .
Secondly, if A is reducible, without loss of generality, we may assume that A has the following block upper triangular form:
A = [ A 11 A 12 A 1 K 0 A 22 A 2 K 0 0 0 0 0 A K K ] ,

where A i i M n i is an irreducible diagonal block matrix, i = 1 , 2 , , K . Obviously, τ ( A A 1 ) = min i τ ( A i i A i i 1 ) . Thus the reducible case is converted into the irreducible case. This proof is completed. □

Theorem 3.2 If A = ( a i j ) M n is a strictly diagonally dominant by row, then
min i { a i i u i R i 1 + j i u j i } min i { a i i s i R i 1 + j i s j i } .
Proof Since A is strictly diagonally dominant by row, for any j i , we have
d j m j i = | a j i | + k j , i | a j k | a j j | a j i | + k j , i | a j k | r i a j j = ( 1 r i ) k j , i | a j k | a j j 0 ,
or equivalently,
d j m j i , j i , j N .
(1)
So, we can obtain
u j i = | a j i | + k j , i | a j k | m k i a j j | a j i | + k j , i | a j k | d k a j j = s j i , j i , j N ,
(2)
and
u i s i , i N .
Therefore, it is easy to obtain that
a i i u i R i 1 + j i u j i a i i s i R i 1 + j i s j i , i N .
Obviously, we have the desired result
min i { a i i u i R i 1 + j i u j i } min i { a i i s i R i 1 + j i s j i } .

This proof is completed. □

Theorem 3.3 If A = ( a i j ) M n is strictly diagonally dominant by row, then
min i { a i i u i R i 1 + j i u j i } min i { a i i m i R i 1 + j i m j i } .
Proof Since A is strictly diagonally dominant by row, for any j i , we have
r i m j i = r i | a j i | + k j , i | a j k | r i a j j = r i | a j i | k j , i | a j k | a j j = r i ( a j j k j , i | a j k | ) | a j i | a j j = a j j k j , i | a j k | a j j ( r i | a j i | a j j k j , i | a j k | ) 0 ,
i.e.,
r i m j i , j i , j N .
(3)
So, we can obtain
u j i = | a j i | + k j , i | a j k | m k i a j j | a j i | + k j , i | a j k | r i a j j = m j i , j i , j N ,
(4)
and
u i m i , i N .
Therefore, it is easy to obtain that
a i i u i R i 1 + j i u j i a i i m i R i 1 + j i m j i , i N .
Obviously, we have the desired result
min i { a i i u i R i 1 + j i u j i } min i { a i i m i R i 1 + j i m j i } .

 □

Remark 3.1 According to inequalities (1) and (3), it is easy to know that
b j i | a j i | + k j , i | a j k | m k i a j j b i i | a j i | + k j , i | a j k | d k a j j b i i , i N .
and
b j i | a j i | + k j , i | a j k | m k i a j j b i i | a j i | + k j , i | a j k | r i a j j b i i , i N .

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 3.4 If A M n is strictly diagonally dominant by row, then
τ ( A A 1 ) min i { 1 1 a i i j i | a j i | u j i } .

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
1 1 a i i j i | a j i | u j i 1 1 a i i j i | a j i | s j i ,
and
1 1 a i i j i | a j i | u j i 1 1 a i i j i | a j i | m j i .

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 = [ 4 1 1 1 2 5 1 1 0 2 4 1 1 1 1 4 ] .
  1. 1.
    Estimate the upper bounds for entries of A 1 = ( b i j ) . Firstly, by Lemma 2.2(2) in [9], we have
    A 1 [ 1 0.5833 0.5000 0.5000 0.6667 1 0.5000 0.5000 0.5000 0.6667 1 0.5000 0.5833 0.5833 0.5000 1 ] [ b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 ] .
     
By Lemma 2.2, we have
A 1 [ 1 0.5625 0.5000 0.5000 0.6167 1 0.5000 0.5000 0.4792 0.6458 1 0.5000 0.5417 0.5625 0.5000 1 ] [ b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 ] .
By Lemma 2.3 and Theorem 3.1 in [9], we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equay_HTML.gif
By Lemma 2.3 and Lemma 3.1, we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equaz_HTML.gif
  1. 2.

    Lower bounds for τ ( A A 1 ) .

     
By Theorem 3.2 in [9], we obtain
0.9755 = τ ( A A 1 ) 0.8000 .
By Theorem 3.1, we obtain
0.9755 = τ ( A A 1 ) 0.8250 .

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Electronic Science and Technology of China

References

  1. Berman A, Plemmons RJ Classics in Applied Mathematics 9. In Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia; 1994.View ArticleGoogle Scholar
  2. Horn RA, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge; 1991.MATHView ArticleGoogle Scholar
  3. Fiedler M, Markham TL: An inequality for the Hadamard product of an M -matrix and inverse M -matrix. Linear Algebra Appl. 1988, 101: 1–8.MATHMathSciNetView ArticleGoogle Scholar
  4. Fiedler M, Johnson CR, Markham T, Neumann M: A trace inequality for M -matrices and the symmetrizability of a real matrix by a positive diagonal matrix. Linear Algebra Appl. 1985, 71: 81–94.MATHMathSciNetView ArticleGoogle Scholar
  5. Song YZ: On an inequality for the Hadamard product of an M -matrix and its inverse. Linear Algebra Appl. 2000, 305: 99–105. 10.1016/S0024-3795(99)00224-4MATHMathSciNetView ArticleGoogle Scholar
  6. Yong XR: Proof of a conjecture of Fiedler and Markham. Linear Algebra Appl. 2000, 320: 167–171. 10.1016/S0024-3795(00)00211-1MATHMathSciNetView ArticleGoogle Scholar
  7. Chen SC: A lower bound for the minimum eigenvalue of the Hadamard product of matrix. Linear Algebra Appl. 2004, 378: 159–166.MATHMathSciNetView ArticleGoogle Scholar
  8. Li HB, Huang TZ, Shen SQ, Li H: Lower bounds for the eigenvalue of Hadamard product of an M -matrix and its inverse. Linear Algebra Appl. 2007, 420: 235–247. 10.1016/j.laa.2006.07.008MATHMathSciNetView ArticleGoogle Scholar
  9. Li YT, Chen FB, Wang DF: New lower bounds on eigenvalue of the Hadamard product of an M -matrix and its inverse. Linear Algebra Appl. 2009, 430: 1423–1431. 10.1016/j.laa.2008.11.002MATHMathSciNetView ArticleGoogle Scholar
  10. Varga RS: Minimal Gerschgorin sets. Pac. J. Math. 1965, 15(2):719–729. 10.2140/pjm.1965.15.719MATHView ArticleGoogle Scholar
  11. Sinkhorn R: A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 1964, 35: 876–879. 10.1214/aoms/1177703591MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Cheng et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. 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.