Two inequalities for the Hadamard product of matrices

Let M _n denote the set of all n × n complex matrices and N denote the set {1, 2, ..., n}. Let A = (a _ij ), B = (b _ij ) ∈ M _n . If a _ij - b _ij ≥ 0, we say that A ≥ B, and if a _ij ≥ 0, we say that A is nonnegative. The spectral radius of A is denoted by ρ(A). If A is a nonnegative matrix, the Perron-Frobenius theorem guarantees that ρ(A) ∈ σ(A), where σ(A) denotes the spectrum of A.

If there does not exist a permutation matrix P such that

where A₁, A₂ are square matrices, then A is called irreducible. Let A be an irreducible nonnegative matrix. It is well known that there exists a positive vector u such that Au = ρ(A)u. The Hadamard product of A, B is defined as A ○ B = (a _ij b _ij ) ∈ M _n . Let A ∈ M _n , and let

Let ς(A) represent the set of all simple circuits in the digraph Γ(A) of A. Recall that a circuit of length k in Γ(A) is an ordered sequence γ = (i₁, ..., i _k , i_k+1), where i₁, ..., i _k ∈ N are all distinct, i_k+1= i₁. The set {i₁, ..., i _k } is called the support of γ and is denoted by γ ̄. The length of the circuit is denoted by |γ|.

In [1], there is a simple estimate for ρ(A ○ B): if A ≥ 0, B ≥ 0, then ρ(A ○ B) ≤ ρ(A)ρ(B).

Recently, using the Gersgorin theorem that involves only elements in one row or column of the matrix, Fang [2] and Huang [3] gave new estimates for ρ(A ○ B) that were better than the result of [1]. Using the Brauer theorem that involves elements in two rows of the matrix at a time, the authors of [4, 5] derived new upper bounds for ρ(A ○ B) that improved the results of [2, 3]. As we all know, besides Gersgorin theorem and Brauer theorem, Brualdi theorem is also an important eigenvalue inclusion theorem and it involves more elements of the matrix than the other two theorems. In view of this, Liu [4] proposed the following problem: Could we get some new estimate better than the previous results using Brualdi theorem? In this paper, we give affirmative conclusions. Two new upper bounds for ρ(A ○ B) are provided. These bounds improve some existing results and numerical examples illustrate that our results are superior.

First, we give some lemmas which are useful for obtaining the main results.

Lemma 2.1 [6] Let A ∈ M _n be a nonnegative matrix. If A _k is a principal submatrix of A, then ρ(A _k ) ≤ ρ(A). If A is irreducible and A _k ≠ A, then ρ(A _k ) < ρ(A).

Lemma 2.2 [7] Let A ∈ M _n be a nonnegative matrix, and let ς(A) ≠ ∅. Then for any diagonal matrix D with positive diagonal entries, we have

Lemma 2.3 [4] Let A, B ∈ M _n . If E; F are diagonal matrices of order n, then

Theorem 2.1 Let A, B ∈ M _n , and A ≥ 0, B ≥ 0. Then

Proof. If A ○ B is irreducible, then A and B are irreducible. From Lemma 2.1, we have

Since A = (a _ij ), B = (b _ij ) are nonnegative irreducible, there exist two positive vectors u, v such that Au = ρ(A)u, Bv = ρ(B)v. Thus, we have

and

Define U = diag(u₁, ..., u _n ), V = diag(v₁, ..., v _n ). Let  = (  i j ) = U - 1 A U, B ^ = ( B ^ i j ) = V - 1 B V. From (2) and (3), we have

and

Let D = VU. According to Lemma 2.2, for the positive diagonal matrix D, we have

Using Lemma 2.3, we have

Then,

So, we have

If A ○ B is reducible, then one of A and B is reducible. If we denote by P = (p _ij ) the n × n permutation matrix with p₁₂ = p₂₃ = · · · = p_{n 1}= 1, the remaining p _ij = 0, then both A + tP and B + tP are nonnegative irreducible matrices for any chosen positive real numbers t. Now, we substitute A + tP and B+tP for A and B, respectively in the previous case, and then letting t → 0, the result follows by continuity.

Two bounds for ρ(A ○ B) given in [2] and [4], respectively, are

and

Next, we give a simple comparison between (1) and (4). It is easy to see

Then the bound (1) is better than the bound (4). From the difference between (1) and (5), we could not verify that (1) is better than (5) in theoretical analysis, but the following numerical example shows that the result derived in Theorem 2.1 is better than (4) and (5).

Example 2.1. Consider two 4 × 4 nonnegative matrices

It is easy to calculate that ρ(A ○ B) = ρ(A) = 5.4983. By inequalities (4) and (5), we have

and ρ(A ○ B) = 11.6478, and by Theorem 2.1, we get

Next, we will give the second inequality for ρ(A ○ B). For A ≥ 0, write L = A - D, where D = diag(a₁₁, ..., a _nn ). with We denote J A = D 1 - 1 L with D₁ = diag(d _ii ), where

Then, J _A is nonnegative, and J _A = A if a _ii = 0 for all i. For B ≥ 0, let D₂ = diag(s _ii ), with

Then the nonnegative matrix J _B can be similarly defined.

Theorem 2.2 Let A, B ∈ M _n , and A ≥ 0, B ≥ 0. Then

Proof. If A ○ B is nonnegative irreducible, then A and B are irreducible. Since J _A and J _B are also nonnegative irreducible, there exist two positive vectors x, y such that J _A x = ρ(J _A )x, J _B y = ρ(J _B )y. So, we have

Let à = ( à i j ) = Ũ - 1 A Ũ, and B ̃ = ( b ̃ i j ) = Ṽ - 1 B Ṽ in which and are nonsingular diagonal matrices Ũ = diag ( x 1 , ⋯ , x n ) and Ṽ = diag ( y 1 , ⋯ , y n ).

From Lemma 2.3, we have

and then

Let W = Ṽ Ũ. Then for the positive diagonal matrix W, it follows from Lemma 2.2 that

If A ○ B is reducible, then substituting A + tP and B + tP for A and B, respectively in the previous case, letting t → 0, the result is derived.

The bounds for ρ(A ○ B) obtained in [3] and [5], respectively, are

and

It can be easily verified that the bound (6) is better than the bound (7). Here too, we could not give the comparison between (6) and (8), but the following example shows that the result obtained in Theorem 2.2 is better than (7) and (8).

Example 2.2. Let

Then

It is clear that ρ(J _A ) = 0.8182, ρ(J _B ) = 1.1258, and ρ(A ○ B) = 6.3365. By (7) and (8), we have

and ρ(A ○ B) = 9.6221, and by Theorem 2.2, we get

In this paper, we propose two new upper bounds for the Hadamard product of matrices. These bounds are better than the results of [2, 3] and numerical examples illustrate that our results are superior than the previous results of [2‐5].