Some new inequalities for nonnegative matrices involving Schur product

: In this study, we focused on the spectral radius of the Schur product. Two new types of the upper bound of ( ) M N  , which is the spectral radius of the Schur product of two matrices , M N with nonnegative elements, were established using the Hölder inequality and eigenvalue inclusion theorem. In addition, the obtained new type upper bounds were compared with the classical conclusions. Numerical examples demonstrated that the new type of upper formulas improved the result of Johnson and Horn effectively in some cases, and were sharper than other existing results.


Introduction
For convenience, we use nn C  to represent the union of complex matrices of order , n nn R  denotes the union of all real matrices of order n and n R denotes the set of vectors of order .n Given a component-wise nonnegative matrix   ( 2 n  ), then M is considered a reducible matrix if there exists a permutation matrix P such that 11 12 T

 =  
, where 11 M and 22 M are square matrices of an order of at least one.Otherwise, M is referred to be irreducible.
If the matrix M is nonnegative, i.e., whose elements are nonnegative, then the spectral radius, represented by ( ) product MN in absolute value.The study of the Schur product, especially the spectral radius of the Schur product, has attracted the attention of a wide range of scholars.Many studies involving the bound of ( )

MN
 can be found in the subsequent works [2,3].
Let , MN be nonnegative matrices.The following classical result can be found in [1], The above-mentioned inequality shows that the spectral radius ( ) MN  is not more than the product of ( ) The improved result of inequality (1.1) was proposed in [4] as follows: In some cases, the two results mentioned above may be fragile.The following example illustrates this situation.Let , MI = the identity matrix of order n and , NJ = the matrix of all ones with the order .
n It is not difficult to observe that We know that these inequalities can be weak when n is very large.The following result is observed owing to Fang [5]: Liu et al [6] improved inequality (1.3) and derived the following conclusion: In addition, some significant new boundaries of the spectral radius of the Schur product were introduced in [7-10], which gave better estimations for the spectral radius in some cases.Inspired by the research, we continue to study the upper bound on the spectral radius of the Schur product of two nonnegative matrices.We provided two new types of the upper bound of the spectral radius involving the Schur product using the eigenvalue inclusion theorem and the Hölder inequality.Numerical tests validate that the new type of upper formulas improve the result of Johnson and Horn [1] effectively in some cases and are sharper than other existing results, which approach the real value more efficiently than previous ones.
pq += , then the following is observed: and a nonnegative nonzero vector M there exists a pair of positive integers ( ) , ij satisfying the following inequality: The main findings of this study are presented below.
are two nonnegative matrices and 1, 1 pq   with 11 1 pq += , then we have the following: (2.1) Proof.When 1 n = , inequality (2.1) becomes an equality, and we assume that 2 n  .To demonstrate this problem, we will distinguish two cases.Case 1. First, we assume that MN is irreducible, then M and N are irreducible, implying that for any Nn = are nonnegative and irreducible.
According to Lemma 1, there exists and where Therefore, we get the following: , Based on Lemma 3, there exists some i such that ( ) .
Case 2. Now, we consider the matrix MN to be reducible.At this point, there is a permutation matrix   Using continuity theory and combining it with Case 1, we can achieve our desired result.Thus, the proof of Theorem 1 is done.
In Theorem 1, when 2 pq ==, we will obtain the following result.
are two nonnegative matrices, then we have the following: Remark 1.For any 1, 2, , in = , we have the following: .
ii ii Therefore, we obtain the following: This implies that In terms of Theorem 2, inequalities (1.2) and (2.7) and noticing that ( ) , we get the following: Therefore, the bound in (2.6) is sharper than ( ) ( ) MN  known in [1].Now, we give an example to illustrate our conclusion.We consider again the numerical example in the introduction.If M is an identity matrix of order n and N denotes the matrix of all ones with the order , n then we obtain the following: It is surprising to see that our bound is the actual value of the spectral radius.
Next, we establish the second inequality for ( ).
Proof.The conclusion holds with equality when 1. n = Next, we assume that 2 n  .To demonstrate this problem, we will discuss two cases.Case 1. First, we suppose that MN is irreducible, then M and N are irreducible.Clearly, for any Nn = are nonnegative and irreducible.
Next, we define two nonsingular positive diagonal matrices as follows: ( ) Let .
From Lemmas 2 and 4, as well as Eqs (2.9) and (2.10), for any 1, 1 pq   with 11 1 pq += , there exists a pair ( ) ( ) Furthermore, Lemma 5 denotes that From inequality (2.11), we obtain the following: As a result, the bound in (2.8) of Theorem 3 is more precise than that in (2.1) of Theorem 1.When 2 pq == in Theorem 3, we get the following conclusion.
are two nonnegative matrices, then we have the following: According to the previous proofs, we have the following conclusions: If we apply Theorem 4 in this study, we will get Then, the Schur product will be as follows:  , we write the following: remaining elements are zero, where MP  + , NP  + are nonnegative and irreducible for any sufficiently small 0.

Example 2 .
Now, we present the second example and look at the following two 44  nonnegative matrices is a characteristic root of M and any other characteristic root  is not more Estimating the spectral radius is widely used in numerical analysis, graph theory, stability theory and other related fields, and it is a relatively active topic in matrix theory research.The Schur product, often called the Hadamard product, is the consequence of multiplying two matrices element by element to create a new matrix.The Schur product of M  in absolute value.= (see [1, Definition 5.0.1]).It is worth noting that the two multiplied matrices must have the same number of rows and columns.Schur product plays an essential role in the matrix theory.It can be used in various applications, including replacing matrix multiplication, blind signal separation, feature selection and image processing.It is obvious that if , MN are two nonnegative matrices, the Schur product MN is nonnegative and the spectral radius ( ) MN  dominates any other characteristic roots of the Schur