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A new inequality on the minimum eigenvalue for the Fan product of nonsingular M-matrices is given. In addition, a new inequality on the spectral radius of the Hadamard product of nonnegative matrices is also obtained. These inequalities can improve considerably some previous results.

Let

We denote by Z_{n} the class of all n × n real matrices, all of whose off-diagonal entries are nonpositive. A matrix _{n} the set of nonsingular M-matrices. We define

The Fan product of two matrices

If

If_{A} is nonnegative. Recently, some authors gave some lower bounds of

The bound of (1) is better than the bound

In [

where

For a nonnegative matrix

The Hadamard product of two matrices

1) If

2) If _{0}, but

3) If

4) If _{0}, j_{0}, then the upper bound of

The bound of

In [

1) If

where

2) If

3) If

4) If _{0}, j_{0}, then the upper bound of

The bound of

The paper is organized as follows. In Section 2, we give a new lower bound of

In this section, we will give a new lower bound of

If

Lemma 1. [

Lemma 2. [

Theorem 1. Let

where

It is evident that the Theorem holds with equality for n = 1. Next, we assume that

(1) First, we assume that _{A} and J_{B} are also irreducible and nonnegative, so

positive vectors

Then we have

Let

It is easy to see that

Thus, we obtain

We next consider the minimum eigenvalue

By Hölder’s inequality, we have

Then, we have

Since

Hence,

i.e.,

(2) Now, assume that _{n} is a nonsingular M-matrix if and only if all its leading principal minors are positive (see [_{ij} zero, then both

Remark 1. By Lemma 2, the bound in Theorem 1 is better than that in Theorem 4 of [

Example 1. Let

By calculating with Matlab 7.1, it is easy to show that

Applying Theorem 4 of [

The numerical example shows that the bound in Theorem 1 is better than that in Theorem 4 of [

In this section, we will give a new upper bound of

Note that

Similarly, the nonnegative matrix

Lemma 3. [

Lemma 4. [

Theorem 2. Let

1) If

where

2) If

3) If

4) If _{0}, j_{0}, then the upper bound of

Proof. It is evident that 4) holds with equality for n = 1. Next, we assume that

(1) First, we assume that

Then we have

Let

It is easy to see that

Thus, we obtain

We next consider the minimum eigenvalue

and

Thus, we obtain

1) If

2) If _{0}, j_{0}, but

3) If

4) If _{0}, j_{0}, then the upper bound of

(2) Now, we assume that _{ij} = 0, then both

Remark 2. By Lemma 2, the bound in Theorem 2 is better than that in Theorem 6 of [

Example 2. Let

By calculation with Matlab 7.1, we have

If we apply Theorem 6 of [

The numerical example shows that the bound in Theorem 2 is better than that in Theorem 6 of [

DongjieGao, (2015) Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices. Advances in Linear Algebra & Matrix Theory,05,90-97. doi: 10.4236/alamt.2015.53009