New lower bounds of the minimum eigenvalue for the Fan product of several M-matrices

1.   Introduction

For convenience, this study adopts the following notations. We employ ${{R}^{n\times n}}$ $\left({{C}^{n\times n}} \right)$ to represent the space of real (complex) matrices with dimension $n$.

Let $A\in {{R}^{n\times n}}$ ($n\ge 2$). Then, $A$ is defined as a reducible matrix if there exists a permutation matrix $P$ such that

where ${{A}_{11}}\in {{R}^{l\times l}}, \, {{A}_{12}}\in {{R}^{l\times \left(n-l \right)}}, \, {{A}_{22}}\in {{R}^{\left(n-l \right)\times \left(n-l \right)}}$ and $O$ is an $\left(n-l \right)\times l$ zero matrix with $1\le l\le n-1$. Otherwise, $A$ is termed irreducible.

Let ${{Z}_{n}}$ denote the collection of real matrices of order $n$, whose non-diagonal elements are nonpositive. If $A\in {{Z}_{n}}$, $A$ is referred to a Z-matrix. It is evident that a sufficient condition for $A\in {{Z}_{n}}$ is that $A$ can be written as:

where $s$ is a real number and the elements of the matrix $P$ are nonnegative. For $A\in {{Z}_{n}}$, let us denote

where $\lambda$ is the characteristic root of the matrix $A$, with $\tau \left(A \right)$ being the minimum eigenvalue of $A$.

If we restrict $s > \rho (P)$ in (1.1), where $\rho (P)$ is the greatest module of the characteristic root of the matrix $P$, we will obtain a special class of Z-matrices, namely M-matrices (see Lemma 2.5.2.1 in ^[1]). The set of nonsingular M-matrices is denoted by ${{M}_{n}}$.

Notably, if $A\in {{M}_{n}}$, then $\tau \left(A \right)$ is a characteristic root of the matrix $A$ (see Problem 19 in Section 2.5 in ^[1]). M-matrices possess several attractive properties and have been extensively studied ^[2,3]. For an M-matrix, research on the minimum eigenvalue holds particular significance and has led to the emergence of numerous new results. In practice, the minimum eigenvalues of the M-matrices can be used to evaluate the stability of a power system. If the absolute value of the minimum eigenvalue is close to zero, it indicates the presence of stability issues in the system. By monitoring and analyzing the minimum eigenvalues of the M-matrices, potential problems in the power system can be detected on time, facilitating the implementation of appropriate measures to improve the stability and reliability of the system.

Unlike the traditional matrix multiplication calculation, the Fan product is a binary operation that takes two matrices of the same dimension and creates a new matrix of the same order. The Fan product of ${{A}_{1}} = \left({{a}_{ij}} \right)\in {{R}^{n\times n}}$ and ${{A}_{2}} = \left({{b}_{ij}} \right)\in {{R}^{n\times n}}$ is denoted by ${{A}_{1}}\star {{A}_{2}} = M = ({{m}_{ij}})$, where

Notably, the two multiplied matrices must have the same structure. For example, let

Then, we have

The Fan product is a fundamental operation in the study of M-matrices. It plays a crucial role in understanding the properties and characteristics of M-matrices. It is used to analyze the interplay between the elements of two M-matrices and study the properties of the resulting matrix, such as eigenvalues, spectral radius and invertibility. Among these studies, the computation and estimation of the minimum eigenvalue of Fan product has become a popular research topic.

Noticeably, if ${{A}_{1}}, {{A}_{2}}$ are M-matrices, then ${{A}_{1}}\star {{A}_{2}}$ and the minimum eigenvalue $\tau \left({{A}_{1}}\star {{A}_{2}} \right)$ is not greater than any other characteristic roots of the Fan product ${{A}_{1}}\star {{A}_{2}}$ in absolute value. Based on the Brauer theorem, Gerschgorin theorem and Brualdi theorem, multiple studies involving the bounds of $\tau \left({{A}_{1}}\star {{A}_{2}} \right)$ were conducted by the authors of ^[4,5,6].

Assuming ${{A}_{1}}, {{A}_{2}}$ as M-matrices, Horn and Johnson ^[1] established the following classical result describing the relationship between $\tau \left({{A}_{1}}\star {{A}_{2}} \right)$ and the product of $\tau \left({{A}_{1}} \right), \tau \left({{A}_{2}} \right)$, that is

Inspired by the definition of the Fan product of two M-matrices, we present the concept of the Fan product of $k$ M-matrices as follows.

Let ${{A}_{1}} = \left({{a}_{ij}} \right), \, {{A}_{2}} = \left({{b}_{ij}} \right), \, \cdots, \, {{A}_{k}} = \left({{k}_{ij}} \right)$ be $n$ by $n$ M-matrices. Define

where

Note that the class of M-matrices is closed under the Fan product and ${{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}}$ is an M-matrix. Therefore, we can generalize inequality (1.2) to any $k$ M-matrices as follows:

Inspired by the research in ^{[4,5,6,7,8,9,10,11,12,13]}, we continue to study the lower bound of $\tau \left({{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}} \right)$. The remainder of this study is organized as follows. First, we present two new types of lower bounds for the minimum eigenvalue involving the Fan product of any $k$ M-matrices ${{A}_{1}}, {{A}_{2}}, \cdots, {{A}_{k}}$ in Section 2. The obtained new bounds generalize some of the previous results. In Section 3, numerical tests are presented to certify our findings and comparisons among these lower bounds are considered.

2.   Two new lower bounds for $\tau \left({{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}} \right)$

To demonstrate our findings, we first introduce some fundamental lemmas. These will be useful in the subsequent proof.

Lemma 1. ^[1] If $A\in {{R}^{n\times n}}$ is an irreducible M-matrix, then

(1) there exists a positive real eigenvalue that is equal to its minimum eigenvalue $\tau \left(A \right)$,

(2) there is an eigenvector $u > 0$ such that $Au = \tau \left(A \right)u$.

Lemma 2. ^[14] Given an irreducible M-matrix $A\in {{R}^{n\times n}}$ and a nonnegative nonzero vector $z\in {{R}^{n}}$, if $Az\ge kz$, then $\tau \left(A \right)\ge k$.

Lemma 3. ^[15] Let $A = \left({{a}_{ij}} \right)\in {{C}^{n\times n}}$ ($n\ge 2$). If $\lambda$ is the characteristic root of the matrix $A$, then there must exist two unequal positive integers $i, j$ that satisfy the following inequality:

where ${{R}_{i}}(A) = \sum\limits_{k\ne i}^{n}{\left| {{a}_{ik}} \right|}$, ${{R}_{j}}(A) = \sum\limits_{k\ne j}^{n}{\left| {{a}_{jk}} \right|}$.

Lemma 4. ^[16] Let ${{x}_{i}}\ge {{y}_{i}}\ge 0$, $i = 1, 2, \cdots, n$. If $\sum\limits_{i = 1}^{n}{\dfrac{1}{{{p}_{i}}}\ge 1}$ with ${{p}_{i}} > 0$, then we have

In the following, we present the first lower bound for $\tau \left({{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}} \right)$.

Theorem 1. Let ${{A}_{1}} = \left({{a}_{ij}} \right), {{A}_{2}} = \left({{b}_{ij}} \right), \cdots, {{A}_{k}} = \left({{k}_{ij}} \right)$ be $n$ by $n$ nonsingular M-matrices. Then,

Proof. Obviously, inequality (2.2) becomes an equality when $n = 1$. We next assume that $n\ge 2$. To demonstrate this problem, let us distinguish two aspects.

Case 1. First, we assume that ${{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}}$ is irreducible. Clearly, ${{A}_{1}}, {{A}_{2}}, \cdots, {{A}_{k}}$ are all irreducible. In addition, we have

Since ${{A}_{1}}, {{A}_{2}}, \cdots, {{A}_{k}}$ are irreducible M-matrices, according to Lemma 1, there exist

satisfying

That is

From the above equations, we have

for all $i\ne j$. Let $z = \left({{z}_{1}}, {{z}_{2}}, \cdots, {{z}_{n}} \right)\in {{R}^{n}}$, in which

We define $A = {{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}}$. For any $i = 1, 2, \cdots, n$, we have

Noticing that

we get

According to Lemma 2, this means that

Case 2. Next, we consider the matrix ${{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}}$ as reducible. As is known, a Z-matrix is a nonsingular M-matrix if and only if all of its leading principal minors are positive (see condition (E17) of Theorem 6.2.3 in ^[14]). At this point, there exists a real number $\varepsilon > 0$ such that ${{A}_{1}}-\varepsilon P$, ${{A}_{2}}-\varepsilon P$, …, ${{A}_{k}}-\varepsilon P$ are irreducible nonsingular M-matrices, where $P = \left({{p}_{ij}} \right)$ is a matrix of size $n$ with

the rest of the elements being zero. If $\varepsilon$ is sufficiently small such that all the leading principal minors of ${{A}_{1}}-\varepsilon P$, ${{A}_{2}}-\varepsilon P$, …, ${{A}_{k}}-\varepsilon P$ are positive, then we replace ${{A}_{1}}-\varepsilon P$, ${{A}_{2}}-\varepsilon P$, …, ${{A}_{k}}-\varepsilon P$ with ${{A}_{1}}, {{A}_{2}}, \cdots, {{A}_{k}}$ in Case 1. Let $\varepsilon \to 0$, we can achieve our desired result by continuity theory. Thus, we have completed the proof of Theorem 1.

Remark 1. We now provide a comparison between the lower bounds in Theorem 1 and the inequality (1.3). In fact, for the nonsingular M-matrices ${{A}_{1}} = \left({{a}_{ij}} \right), {{A}_{2}} = \left({{b}_{ij}} \right), \cdots, {{A}_{k}} = \left({{k}_{ij}} \right)$, we have

It follows from Lemma 4 that

Therefore, we have

This implies that the bound in Theorem 1 is sharper than that in inequality (1.3).

Here, we consider a special case. Let $k = 2$ in Theorem 1, we will obtain the following conclusion.

Corollary 1. Let ${{A}_{1}} = \left({{a}_{ij}} \right), \, {{A}_{2}} = \left({{b}_{ij}} \right)$ be $n$ by $n$ nonsingular M-matrices. Then,

This happens to be the conclusion of Theorem 9 of Fang ^[4]. Therefore, the result of Fang ^[4] is included in Theorem 1 of this paper.

The second inequality regarding $\tau \left({{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}} \right)$ will be established next.

Theorem 2. Let ${{A}_{1}} = \left({{a}_{ij}} \right), \, {{A}_{2}} = \left({{b}_{ij}} \right), \cdots, {{A}_{k}} = \left({{k}_{ij}} \right)$ be $n$ by $n$ nonsingular M-matrices. Then,

Proof. Obviously, the conclusion is true when $n = 1$. Next, we assume that $n\ge 2$. To demonstrate this problem, let us distinguish two aspects.

Case 1. ${{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}}$ is irreducible. It is known that ${{A}_{1}}, {{A}_{2}}, \cdots, {{A}_{k}}$ are all irreducible. According to Lemma 1, there exist

satisfying

Therefore, we have

Now, we define $k$ nonsingular positive diagonal matrices as follows:

Let

and denote ${{\tilde{A}}_{1}}\star {{\tilde{A}}_{2}}\star \cdots \star {{\tilde{A}}_{k}} = H = \left({{h}_{ij}} \right)$. By the definition of the matrix ${{\tilde{A}}_{1}}\star {{\tilde{A}}_{2}}\star \cdots \star {{\tilde{A}}_{k}}$, we have

Define $D = {{D}_{1}}{{D}_{2}}\cdots {{D}_{k}}$ and ${{D}^{-1}}\left({{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}} \right)D = {H}' = \left({{h}_{ij}}^{\prime } \right)$. Thus, we get

Therefore, we have

This shows that

In addition, we have

Similarly, we have

Since $\tau \left({{{\tilde{A}}}_{1}}\star {{{\tilde{A}}}_{2}}\star \cdots \star {{{\tilde{A}}}_{k}} \right)$ is an eigenvalue of ${{\tilde{A}}_{1}}\star {{\tilde{A}}_{2}}\star \cdots \star {{\tilde{A}}_{k}}$, in terms of Lemma 3 and inequalities (2.5) and (2.6), there exist two unequal positive integers $i, j$ such that

From inequality (2.7) and $0 < \tau \left({{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}} \right) < {{a}_{ii}}{{b}_{ii}}\cdots {{k}_{ii}}$ for $i = 1, 2, \cdots, n$, we get

Solving inequality (2.8), we obtain

Case 2. ${{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}}$ is reducible. We can use the approach of Theorem 1 similarly prove this. Hence, the proof of Theorem 2 is finished.

Remark 2. Following the demonstration of Theorem 2, we present a new proof of Theorem 1. From Theorem 1.11 in ^[15], we obtain

According to $0 < \tau \left({{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}} \right)\le {{a}_{ii}}{{b}_{ii}}\cdots {{k}_{ii}}$ and inequality (2.5), we obtain

Thus, we acquire

Now, we consider a special case. We can immediately obtain the following corollary from Theorem 2 by setting $k = 2$.

Corollary 2. Let ${{A}_{1}} = \left({{a}_{ij}} \right), {{A}_{2}} = \left({{b}_{ij}} \right)$ be $n$ by $n$ nonsingular M-matrices. Then,

This happens to be the conclusion of Theorem 7 of Liu ^[5]. Therefore, the result of Liu ^[5] is included in Theorem 2 of this paper.

The following theorem shows that the bound in (2.4) of Theorem 2 is more precise than the bound in (2.2) of Theorem 1.

Theorem 3. Let ${{A}_{1}} = \left({{a}_{ij}} \right), {{A}_{2}} = \left({{b}_{ij}} \right), \cdots, {{A}_{k}} = \left({{k}_{ij}} \right)$ be $n$ by $n$ nonsingular M-matrices. Then,

Proof. We assume, without losing generality, that

As a result, we can express the inequality above in the following way:

Therefore, we have

From inequalities (2.4) and (2.10), we obtain

The proof of Theorem 3 is finished.

Remark 3. From the previous statements, we observe that

3.   Numerical examples

To demonstrate that our new lower bounds are more precise than the previous results, we consider two specific examples in this section.

Example 1. First, we employ two M-matrices from ^[6].

We compute the Fan product:

It is simple to see that $\tau \left({{A}_{1}} \right) = 0.5402$, $\tau \left({{A}_{2}} \right) = 0.3432$ and $\tau \left({{A}_{1}}\star {{A}_{2}} \right) = 0.9377$. By inequality (1.2) in ^[1], we get

According to Corollary 1 (see also Theorem 9 in ^[4]), we have

In terms of Corollary 2 (see also Theorem 7 in ^[5]), we obtain

Example 2. Now, we present the second example and examine the following three M-matrices.

We compute the Fan product:

It is easy to observe that $\tau \left({{A}_{1}} \right) = \tau \left({{A}_{2}} \right) = \tau \left({{A}_{3}} \right) = 1$, $\tau \left({{A}_{1}}\star {{A}_{2}}\star {{A}_{3}} \right) = 100$. By inequality (1.3), we obtain

We observe that this result is trivial. If we apply Theorem 1 in this study, we acquire

Surprisingly, the proposed is the actual minimum eigenvalue of $\tau \left({{A}_{1}}\star {{A}_{2}}\star {{A}_{3}} \right)$. From the presented examples, we can see that our results are more accurate than the earlier results in some cases.

4.   Conclusions

M-matrices are a special class of matrices with important properties. The Fan product is a binary operation defined for M-matrices, which plays an important role in understanding the properties and characteristics of M-matrices. Inspired by the definition of the Fan product of two M-matrices, we introduced the concept of the Fan product of $k$ M-matrices.

Additionally, for M-matrices ${{A}_{1}}, {{A}_{2}}, \cdots, {{A}_{k}}$, we have proposed two new inequalities for the lower bound of the minimum eigenvalue of the Fan product ${{A}_{1}}\star {{A}_{2}}\star \cdots \star {{A}_{k}}$. The derived new type lower bounds generalize some of the existing results to a certain extent.

In summary, this study established the relationship between the minimum eigenvalue of the Fan product of $k$ M-matrices and the minimum eigenvalues of the original $k$ M-matrices. The conclusions of this study can be considered as a valuable addition to the theoretical study of M-matrices.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this paper.

Acknowledgments

The Natural Science Research Project of the Education Department of Sichuan Province (No.18ZB0364) provided financial support for this study.