Lower bounds on the minimum eigenvalue of the Fan product of several M -matrices

The concept of the Fan product of several M -matrices is introduced. Furthermore, two new lower bounds of the minimum eigenvalue of the Fan product of several M -matrices are proposed. These obtained new lower bounds generalize and improve some earlier ﬁndings. One example is presented to illustrate the precision of the given lower bounds


Introduction
Many issues in the social, physical, and biological sciences can be reduced to problems using matrices that possess a unique structure owing to limitations.One of the most common situations is when the matrix K has nonpositive off-diagonal entries.The matrix K can be written as follows: K = sI -P, P ≥ 0. (1.1) Here, P ≥ 0 means that the matrix P is nonnegative.Let R n×n (C n×n ) denote the union of n-by-n real (complex) matrices.Here, the conventional notation is employed by setting The aim is to study a special subclass of matrices in Z n called M-matrices.
For any matrix K of the form in Eq. (1.1), if s > ρ(P), the spectral radius of P, then K is defined to be a nonsingular M-matrix.The set of nonsingular M-matrices is denoted by M n .Let K ∈ M n and assume K = sI -P with s > ρ(P) and P ≥ 0. It is known that q(K) = sρ(P) is an eigenvalue of the matrix K with the minimum module [1], and q(K) is considered to be the minimum eigenvalue of the M-matrix K .
M-matrices have been widely investigated and possess many appealing qualities [2,3].Research on the minimum eigenvalue is particularly important for an M-matrix and has produced many novel findings.In practice, the minimum eigenvalues of the M-matrices play an important role in evaluating the stability of a power system.Potential issues with the power system can be identified early by tracking and examining the minimum eigenvalues of the M-matrices, making it easier to obtain the proper solutions and increase the stability and reliability of the system.
For two matrices A 1 = (a ij ) ∈ R n×n and A 2 = (b ij ) ∈ R n×n , the Fan product of A 1 and A 2 is denoted by A 1 A 2 = (s ij ), where 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.In previous studies, the computation and estimation of the minimum eigenvalue of the Fan product has become a popular research topic, and many results have been obtained [4][5][6]. Let The following classical result is proposed by Horn and Johnson [1]: The above inequality shows that the minimum eigenvalue of the Fan product A 1 A 2 is more than the product of the minimum eigenvalues of A 1 and A 2 .
As the class of M-matrices is closed under the Fan product, one can generalize the definition of the Fan product from two to several M-matrices.
, where From the inequality in Eq. (1.2), one can observe that Motivated by previous work [4][5][6][7][8][9], in this study, the lower bound of q(A 1 A 2 • • • A m ) was investigated further.The structure of the article is as follows.
In Sect.2, a new lower bound on the minimum eigenvalue involving the Fan product of several M-matrices is introduced.In Sect.3, this result is further improved.These new lower bounds generalize some earlier findings.
To verify the conclusions, a numerical test is described in Sect.4, and these lower bounds are compared.

New lower bound for q(
This section begins with a basic definition.Definition 1 Let A ∈ R n×n with n ≥ 2. If there exists a permutation matrix P that satisfies where B and D are square matrices, A is considered reducible; if such a permutation matrix P does not exist, A is irreducible. For the M-matrices A 1 = (a ij ), A 2 = (b ij ), . . ., A m = (m ij ) of order n and k = 1, 2, one can write where and In addition, it is noted that Thus, D 1 , D 2 , . . ., D m are all nonsingular.One can define It is obvious that A m are nonnegative.The following important lemmas must be remembered to arrive at the primary conclusions of this work.
Lemma 1 [10] Let A ∈ R n×n be an irreducible nonnegative matrix.The following facts apply: (1) There is a positive real eigenvalue that equals its spectral radius ρ(A).
Lemma 2 [10] If an irreducible M-matrix A ∈ R n×n and a nonnegative nonzero vector z satisfy Az ≥ kz, then q(A) ≥ k.
Lemma 3 [11] Let Lemma 5 [10] Let A = (a ij ) ∈ C n×n (n ≥ 2).For any eigenvalue λ of the matrix A, there must exist two unequal positive integers i, j satisfying the inequality The first result of the lower bound for (2.1) This problem can be solved in two cases.First, A is considered irreducible.One can then see that A 1 , A 2 , . . ., A m are all irreducible.Therefore, A m are all irreducible and nonnegative for k = 1, 2. From Lemma 1, there exist m vectors x = (x 1 , x 2 , . . ., x n ) T > 0, y = (y 1 , y 2 , . . ., y n ) T > 0, . . ., z = (z 1 , z 2 , . . ., z n ) T > 0 that satisfy . . . . . . where For the irreducible M-matrix A, according to Lemma 3, By Lemma 2 and the inequality in Eq. (2.5), In the following, it is assumed that the matrix A is reducible.Let H = (h ij ) be the n-by-n permutation matrix with the remaining h ij = 0.A sufficiently small positive number ε is chosen such that A 1 -εH, A 2 -εH, . . ., A m -εH are irreducible nonsingular M-matrices.Substituting A 1 -εH, A 2 -εH, . . ., A m -εH for A 1 , A 2 , . . ., A m in the irreducible case, and then by setting ε → 0, the conclusion holds by continuity theory.
Next, two special cases are considered.By setting m = 2, k = 1 in Theorem 1, the conclusion is obtained as follows.
This is the result of Theorem 4 in a previous report [4].Let m = k = 2, then, Theorem 1 yields the following corollary, which is the conclusion of Theorem 2.7 of Li [5].
As a result, the conclusions of Theorem 4 in an earlier report [4] and Theorem 2.7 in other work [5] are contained in Theorem 1 of this study.

Remark 1 From Lemma 4, one can get
This shows that Therefore, the lower bound in the inequality in Eq. (2.7) is superior to the lower bound in the inequality in Eq. (2.6).

Improved lower bound for q(
In this section, a second lower bound is proposed for which is an improvement of the lower bound in Sect. 2. To illustrate this issue, two aspects are considered.First, it is assumed that A is irreducible.One can see that A 1 , A 2 , . . ., A m are all irreducible.In addition, J (k) A 1 , J (k) A 2 , . . ., J (k) A m are all irreducible and nonnegative for k = 1, 2. In terms of Lemma 1, for k = 1, 2, there exist . . . . . . where According to Eqs. (3.2)-(3.4),we arrive at Now, the following is defined: Clearly, P 1 , P 2 , . . ., P m are nonsingular.Let . ., and let From the definition of the Fan product of several M-matrices, one obtains It is assumed that Thus, we have This implies that In addition, according to Lemma 3, one obtains Similarly, one obtains As q( Ã1 Ã2 • • • Ãm ) is an eigenvalue of Ã1 Ã2 • • • Ãm , it follows from Lemma 5 that there exist two unequal positive integers i, j that satisfy Combining the inequalities in Eqs.(3.5) and (3.6), one obtains From the inequality in Eq. (3.8), we acquire

Now, considering that the matrix
reducible, one can prove similarly by following the proof of Theorem 1.
Remark 2 A novel proof of Theorem 1 is introduced.According to the Gerschgorin theorem [10], Combining the inequalities in Eq. (3.5) and

This indicates q(
The following corollary is a special case of Theorem 2 by setting m = 2, k = 1.
This is the direct result of Theorem 2 of Liu [6].Setting m = k = 2 in Theorem 2, one can obtain the following conclusion.
A 2 1 2 .(3.10)This happens to be the conclusion of Theorem 2.8 in an earlier report [5].Therefore, the results of Theorem 2 in another report [6] and Theorem 2.8 in the earlier report [5] are contained in Theorem 2 of this study.Remark 3 According to Lemma 4, This implies that the lower bound in the inequality in Eq. (3.10) is superior to the lower bound in the inequality in Eq. (3.9).Next, the two lower bounds for q(A 1 A 2 • • • A m ) in Theorem 1 and Theorem 2 are compared.
Proof It can be assumed that As a result, the above inequality can be expressed as Then, one obtains which, together with the inequalities in Eqs.(3.1) and (3.11), leads to Therefore, the conclusion is proved.
The result is trivial.One can see from the example provided that, in certain instances, the results are more accurate than earlier results.

Conclusions
For the Fan product of M-matrices A 1 , A 2 , . . ., A m , two new inequalities on the lower bound of q(A 1 A 2 • • • A m ) were proposed.The derived new lower bounds generalize some previous results.

2 + 4 (
b ii c ii ) = 16.9646.Utilizing Theorem 2, one obtainsq(A 1 A 2 A 3 ) ≥ min i =j 1 2 a ii b ii c ii + a jj b jj c jj -(a ii b ii c iia jj b jj c jj ) 2 + 4(a ii b ii c ii )(a jj b jj c jj )ρ 2 (J A 1 )ρ 2 (J A 2 )ρ 2 (J A 3 ) ii c ii + a jj b jj c jj -(a ii b ii c iia jj b jj c jj ) a ii b ii c ii )(a jj b jj c jj )ρ J (2) A 1 ρ J (2)A 2 ρ J(2)