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Research | Open | Published:

Some new inequalities for the Hadamard product of M-matrices

Abstract

If A and B are n×n nonsingular M-matrices, a new lower bound for the minimum eigenvalue τ(B A 1 ) for the Hadamard product of B and A 1 is derived. As a consequence, a new lower bound for the minimum eigenvalue τ(A A 1 ) for the Hadamard product of A and its inverse A 1 is given. Theoretical results and an example demonstrate that the new bounds are better than some existing ones.

MSC:15A06, 15A18, 15A48.

1 Introduction

For convenience, for any positive integer n, let N={1,2,,n} throughout. The set of all n×n real matrices is denoted by R n × n and C n × n denotes the set of all n×n complex matrices.

A matrix A=( a i j ) R n × n is called a nonnegative matrix if a i j 0. The spectral radius of A is denoted by ρ(A). If A is a nonnegative matrix, the Perron-Frobenius theorem guarantees that ρ(A) is an eigenvalue of A.

Z n denotes the class of all n×n real matrices all of whose off-diagonal entries are nonpositive. An n×n matrix A is called an M-matrix if there exists an n×n nonnegative matrix B and a nonnegative real number λ such that A=λIB and λρ(B), I is the identity matrix; if λ>ρ(B), we call A a nonsingular M-matrix; if λ=ρ(B), we call A a singular M-matrix. Denote by M n the set of nonsingular M-matrices.

Let A Z n , and let τ(A)=min{Re(λ):λσ(A)}. Basic for our purpose are the following simple facts (see Problems 16, 19 and 28 in Section 2.5 of [1]):

  1. (1)

    τ(A)σ(A); τ(A) is called the minimum eigenvalue of A.

  2. (2)

    If A,B M n , and AB, then τ(A)τ(B).

  3. (3)

    If A M n , then ρ( A 1 ) is the Perron eigenvalue of the nonnegative matrix A 1 , and τ(A)= 1 ρ ( A 1 ) is a positive real eigenvalue of A.

For two matrices A=( a i j ) and B=( b i j ), the Hadamard product of A and B is the matrix AB=( a i j b i j ). If A and B are two nonsingular M-matrices, then B A 1 is also a nonsingular M-matrix [2].

Let A,B M n and A 1 =( β i j ), in [[1], Theorem 5.7.31] the following classical result is given:

τ ( B A 1 ) τ(B) min 1 i n β i i .
(1.1)

Huang [[3], Theorem 9] improved this result and obtained the following result:

τ ( B A 1 ) 1 ρ ( J A ) ρ ( J B ) 1 + ρ 2 ( J A ) min 1 i n b i i a i i ,
(1.2)

where ρ( J A ), ρ( J B ) are the spectral radii of J A and J B .

The lower bound (1.1) is simple, but not accurate enough. The lower bound (1.2) is difficult to evaluate.

Recently, Li [[4], Theorem 2.1] improved these two results and gave a new lower bound for τ(B A 1 ), that is,

τ ( B A 1 ) min i { b i i s i k i | b k i | a i i } ,
(1.3)

where r l i = | a l i | | a l l | k l , i | a l k | , li; r i = max l i { r l i }, iN; s j i = | a j i | + k j , i | a j k | r k a j j , ji, jN; s i = max j i { s i j }, iN.

For an M-matrix A, Fiedler et al. showed in [5] that τ(A A 1 )1. Subsequently, Fiedler and Markham [[2], Theorem 3] gave a lower bound on τ(A A 1 ),

τ ( A A 1 ) 1 n ,
(1.4)

and proposed the following conjecture:

τ ( A A 1 ) 2 n .
(1.5)

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

Li [[8], Theorem 3.1] obtained the following result:

τ ( A A 1 ) min i { a i i t i R i 1 + j i t j i } ,
(1.6)

which only depends on the entries of A=( a i j ), where R i = k i | a i k |; d i = R i | a i i | , iN; t j i = | a j i | + k j , i | a j k | d k | a j j | , ji, jN; t i = max j i { t i j }, iN.

Li [[9], Theorem 3.2] improved the bound (1.6) and obtained the following result:

τ ( A A 1 ) min i { a i i m i R i 1 + j i m j i } ,
(1.7)

where r l i = | a l i | | a l l | k l , i | a l k | , li; r i = max l i { r l i }, iN; m j i = | a j i | + k j , i | a j k | r i | a j j | , ji, jN; m i = max j i { m i j }, iN.

Recently, Li [[10], Theorem 3.2] improved the bound (1.7) and gave a new lower bound for τ(A A 1 ), that is,

τ ( A A 1 ) min i { a i i T i R i 1 + j i T j i } ,
(1.8)

where T j i =min{ m j i , s j i }, ji; T i = max j i { T i j }, iN.

In the present paper, we present a new lower bound on τ(B A 1 ). As a consequence, we present a new lower bound on τ(A A 1 ). These bounds improve several existing results.

The following is the list of notations that we use throughout: For i,j,k,lN,

R i = k i | a i k | , C i = k i | a k i | , d i = R i | a i i | , c ˆ i = C i | a i i | ; r l i = | a l i | | a l l | k l , i | a l k | , l i ; r i = max l i { r l i } , i N ; c i l = | a i l | | a l l | k l , i | a k l | , l i ; c i = max l i { c i l } , i N ; m j i = | a j i | + k j , i | a j k | r i | a j j | , j i ; m i = max j i { m i j } , i N ; s j i = | a j i | + k j , i | a j k | r k | a j j | , j i ; s i = max j i { s i j } , i N ; T j i = min { m j i , s j i } , j i ; T i = max j i { T i j } , i N .

2 Some lemmas and the main results

In order to prove our results, we first give some lemmas.

Lemma 2.1 [11]

If A=( a i j ) R n × n is an M-matrix, then there exists a diagonal matrix D with positive diagonal entries such that D 1 AD is a strictly row diagonally dominant M-matrix.

Lemma 2.2 [1]

Let A,B=( a i j ) C n × n and suppose that D C n × n and E C n × n are diagonal matrices. Then

D(AB)E=(DAE)B=(DA)(BE)=(AE)(DB)=A(DBE).

Lemma 2.3 [10]

If A=( a i j ) R n × n is a strictly row diagonally dominant M-matrix, then A 1 =( β i j ) satisfies

β j i T j i β i i ,i,jN,ij.

Lemma 2.4 [12]

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

Lemma 2.5 [9]

Let A=( a i j ) R n × n be a strictly row diagonally dominant M-matrix. Then, for A 1 =( β i j ), we have

β i i 1 a i i ,iN.

Lemma 2.6 [10]

If A=( a i j ) R n × n is an M-matrix and A 1 =( β i j ) is a doubly stochastic matrix, then

β i i 1 1 + j i T j i ,iN.

Lemma 2.7 [13]

Let A=( a i j ) C n × n , and let x 1 , x 2 ,, x n be positive real numbers. Then all the eigenvalues of A lie in the region

i , j = 1 i j n { z C : | z a i i | | z a j j | ( x i k i 1 x k | a k i | ) ( x j k j 1 x k | a k j | ) } .

Theorem 2.1 Let A,B=( b i j ) R n × n be two nonsingular M-matrices, and let A 1 =( β i j ). Then

τ ( B A 1 ) min i j 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } .
(2.1)

Proof It is evident that (2.1) is an equality for n=1.

We next assume that n2.

If A is an M-matrix, then by Lemma 2.1 we know that there exists a diagonal matrix D with positive diagonal entries such that D 1 AD is a strictly row diagonally dominant M-matrix and satisfies

τ ( B A 1 ) =τ ( D 1 ( B A 1 ) D ) =τ ( B ( D 1 A D ) 1 ) .

So, for convenience and without loss of generality, we assume that A is a strictly row diagonally dominant M-matrix. Therefore, 0< T i <1, iN.

If B A 1 is irreducible, then B and A are irreducible. Let τ(B A 1 )=λ, so that 0<λ< b i i β i i , iN. Thus, by Lemma 2.7, there is a pair (i,j) of positive integers with ij such that

|λ b i i β i i ||λ b j j β j j | ( T i k i 1 T k | b k i β k i | ) ( T j k j 1 T k | b k j β k j | ) .

Observe that

( T i k i 1 T k | b k i β k i | ) ( T j k j 1 T k | b k j β k j | ) ( T i k i 1 T k | b k i | T k i β i i ) ( T j k j 1 T k | b k j | T k j β j j ) ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) .

Thus, we have

|λ b i i β i i ||λ b j j β j j | ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) .

Then we have

λ 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } .

That is,

τ ( B A 1 ) 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } min i j 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } .

Now, assume that B A 1 is reducible. It is known that a matrix in Z n 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 [14]). If we denote by D=( d i j ) the n×n permutation matrix with d 12 = d 23 == d n 1 , n = d n 1 =1, then both AtD and BtD are irreducible nonsingular M-matrices for any chosen positive real number t, sufficiently small such that all the leading principal minors of both AtD and BtD are positive. Now we substitute AtD and BtD for A and B, respectively in the previous case, and then letting t0, the result follows by continuity. □

Theorem 2.2 Let A,B=( b i j ) R n × n be two nonsingular M-matrices, and let A 1 =( β i j ). Then

min i j 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } min 1 i n { b i i s i k i | b k i | a i i } .

Proof Since T j i =min{ m j i , s j i }, ji, T i = max j i { T i j }, so T i s i , iN. Without loss of generality, for ij, assume that

b i i β i i T i k i | b k i | β i i b j j β j j T j k j | b k j | β j j .
(2.2)

Thus, (2.2) is equivalent to

T j k j | b k j | β j j T i k i | b k i | β i i + b j j β j j b i i β i i .
(2.3)

From (2.1) and (2.3), we have

1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T i k i | b k i | β i i + b j j β j j b i i β i i ) ] 1 2 } = 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) 2 + 4 ( T i k i | b k i | β i i ) ( b j j β j j b i i β i i ) ] 1 2 } = 1 2 { b i i β i i + b j j β j j [ ( b j j β j j b i i β i i + 2 T i k i | b k i | β i i ) 2 ] 1 2 } = 1 2 { b i i β i i + b j j β j j ( b j j β j j b i i β i i + 2 T i k i | b k i | β i i ) } = b i i β i i T i k i | b k i | β i i = β i i ( b i i T i k i | b k i | ) β i i ( b i i s i k i | b k i | ) b i i s i k i | b k i | a i i .

Thus, we have

τ ( B A 1 ) min i j 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } min 1 i n { b i i s i k i | b k i | a i i } .

This proof is completed. □

Remark 2.1 Theorem 2.2 shows that the result of Theorem 2.1 is better than the result of Theorem 2.1 in [4].

If A=B, according to Theorem 2.1, we can obtain the following corollary.

Corollary 2.1 Let A=( a i j ) R n × n be an M-matrix, and let A 1 =( β i j ) be a doubly stochastic matrix. Then

τ ( A A 1 ) min i j 1 2 { a i i β i i + a j j β j j [ ( a i i β i i a j j β j j ) 2 + 4 ( T i k i | a k i | β i i ) ( T j k j | a k j | β j j ) ] 1 2 } .
(2.4)

Theorem 2.3 Let A=( a i j ) R n × n be an M-matrix, and let A 1 =( β i j ) be a doubly stochastic matrix. Then

min i j 1 2 { a i i β i i + a j j β j j [ ( a i i β i i a j j β j j ) 2 + 4 ( T i k i | a k i | β i i ) ( T j k j | a k j | β j j ) ] 1 2 } min i { a i i T i R i 1 + j i T j i } .

Proof Since A is an irreducible M-matrix and A 1 is a doubly stochastic matrix by Lemma 2.4, we have

a i i = k i | a i k |+1= k i | a k i |+1,iN.

Without loss of generality, for ij, assume that

a i i β i i T i k i | a k i | β i i a j j β j j T j k j | a k j | β j j .
(2.5)

Thus, (2.5) is equivalent to

T j k j | a k j | β j j a j j β j j a i i β i i + T i k i | a k i | β i i .
(2.6)

From (2.4) and (2.6), we have

1 2 { a i i β i i + a j j β j j [ ( a i i β i i a j j β j j ) 2 + 4 ( T i k i | a k i | β i i ) ( T j k j | a k j | β j j ) ] 1 2 } 1 2 { a i i β i i + a j j β j j [ ( a i i β i i a j j β j j ) 2 + 4 ( T i k i | a k i | β i i ) ( a j j β j j a i i β i i + T i k i | a k i | β i i ) ] 1 2 } = 1 2 { a i i β i i + a j j β j j [ ( a i i β i i a j j β j j ) 2 + 4 ( T i k i | a k i | β i i ) 2 + 4 ( T i k i | a k i | β i i ) ( a j j β j j a i i β i i ) ] 1 2 } = 1 2 { a i i β i i + a j j β j j [ ( a j j β j j a i i β i i + 2 T i k i | a k i | β i i ) 2 ] 1 2 } = 1 2 { a i i β i i + a j j β j j ( a j j β j j a i i β i i + 2 T i k i | a k i | β i i ) } = a i i β i i T i k i | a k i | β i i = β i i ( a i i T i k i | a k i | ) a i i T i R i 1 + j i T j i .

Thus, we have

τ ( A A 1 ) min i j 1 2 { a i i β i i + a j j β j j [ ( a i i β i i a j j β j j ) 2 + 4 ( T i k i | a k i | β i i ) ( T j k j | a k j | β j j ) ] 1 2 } min i { a i i T i R i 1 + j i T j i } .

This proof is completed. □

Remark 2.2 Theorem 2.3 shows that the result of Corollary 2.1 is better than the result of Theorem 3.2 in [10].

3 Example

For convenience, we consider that the M-matrices A and B are the same as the matrices of [4].

A=[ 4 1 1 1 2 5 1 1 0 2 4 1 1 1 1 4 ],B=[ 1 0.5 0 0 0.5 1 0.5 0 0 0.5 1 0.5 0 0 0.5 1 ].
  1. (1)

    We consider the lower bound for τ(B A 1 ).

If we apply (1.1), we have

τ ( B A 1 ) τ(B) min 1 i n β i i =0.07.

If we apply (1.2), we have

τ ( B A 1 ) 1 ρ ( J A ) ρ ( J B ) 1 + ρ 2 ( J A ) min 1 i n b i i a i i =0.048.

If we apply (1.3), we have

τ ( B A 1 ) min i { b i i s i k i | b k i | a i i } =0.08.

If we apply Theorem 2.1, we have

τ ( B A 1 ) min i j 1 2 { b i i β i i + b j j β j j [ ( b i i β i i b j j β j j ) 2 + 4 ( T i k i | b k i | β i i ) ( T j k j | b k j | β j j ) ] 1 2 } = 0.1753 .

In fact, τ(B A 1 )=0.2148.

  1. (2)

    We consider the lower bound for τ(A A 1 ).

If we apply (1.5), we have

τ ( A A 1 ) 2 n = 1 2 =0.5.

If we apply (1.6), we have

τ ( A A 1 ) min i { a i i t i R i 1 + j i t j i } =0.6624.

If we apply (1.7), we have

τ ( A A 1 ) min i { a i i m i R i 1 + j i m j i } =0.7999.

If we apply (1.8), we have

τ ( A A 1 ) min i { a i i T i R i 1 + j i T j i } =0.85.

If we apply Corollary 2.1, we have

τ ( A A 1 ) min i j 1 2 { a i i β i i + a j j β j j [ ( a i i β i i a j j β j j ) 2 + 4 ( T i k i | a k i | β i i ) ( T j k j | a k j | β j j ) ] 1 2 } = 0.9755 .

In fact, τ(A A 1 )=0.9755.

Remark 3.1 The numerical example shows that the bounds of Theorem 2.1 and Corollary 2.1 are sharper than those of Theorem 2.1 in [4] and Theorem 3.2 in [10].

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Acknowledgements

This work is supported by the Natural Science Foundation of China (No: 71161020).

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Correspondence to Fu-bin Chen.

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Keywords

  • M-matrix
  • lower bounds
  • Hadamard product
  • minimum eigenvalue