The Hadamard Product of a Nonsingular General H-Matrix and Its Inverse Transpose Is Diagonally Dominant

Abstract

We study the combined matrix of a nonsingular H-matrix. These matrices can belong to two different H-matrices classes: the most common, invertible class, and one particular class named mixed class. Different results regarding diagonal dominance of the inverse matrix and the combined matrix of a nonsingular H-matrix belonging to the referred classes are obtained. We conclude that the combined matrix of a nonsingular H-matrix is always diagonally dominant and then it is an H-matrix. In particular, the combined matrix in the invertible class remains in the same class.

1. Introduction

The Hadamard product of a nonsingular general H-matrix and its inverse transpose, that is, the combined matrix, has been studied in several works such as [1–3]. A complete study of the combined matrix showing its linear application can be seen in [4]. In the last decade, new properties of the combined matrix have been presented in [5, 6]. It is in this last reference where the name of combined matrix appears for the first time.

It is well known that the sum by row or by column of the entries of the combined matrix of a nonsingular matrix , , is exactly equal to 1. Then, if , the combined matrix is doubly stochastic. In [7, 8], the authors studied conditions under which the combined matrix of some classes of matrices is nonnegative. In particular, the authors have studied the nonnegativity of the combined matrix of totally positive (nonnegative) and totally negative (nonpositive) matrices and of sign regular matrices.

The combined matrix has different applications. In a process control problem, if represents the relation among inputs and outputs, the combined matrix of represents the relative gain array of the process. This interpretation was given in [9] and was applied in chemistry, for instance, in [10]. In mathematics, the combined matrix of is used in [11] to compute the eigenvalues of .

Results involving the Hadamard product of H-matrices can be found in [12, 13]. The result where the combined matrix of a nonsingular M-matrix is also a nonsingular M-matrix was obtained by Fiedler in [1]. The same statement can be deduced from [14, 15] as Fiedler and Markham indicated in [3]. In this work we extend this result to nonsingular H-matrices. Firstly, we recall nonsingular H-matrices properties and their relations with diagonal dominance. In Section 3.1, it is proven that the combined matrix of an H-matrix of the invertible class is also an H-matrix of this class. Moreover, in Section 3.2, we obtain properties on diagonal dominance of the inverse matrix and the combined matrix of a nonsingular H-matrix belonging to the mixed class. So, we conclude that the combined matrix of a nonsingular H-matrix of the mixed class is also an H-matrix.

2. Notations and Definitions

In this paper, we work with square matrices of size . Matrices we are considering are real or complex. We will point out when we assume real matrices.

Let the set of indices . Given a matrix of order , the symbol denotes the principal submatrix of order that results after deleting row and column of , whereas the symbol denotes its determinant, .

In order to introduce the concept of combined matrix, we recall that represents the Hadamard product or entrywise product of the matrices and :

Definition 1. The combined matrix of the matrix is defined as

As a consequence, the entry of is given by according to the previous notation.

A class of matrices that allows us to obtain interesting properties of their combined matrices is the class of irreducible matrices. We recall that a matrix is reducible if there exists a permutation matrix such that is block triangular. That is, where the diagonal blocks and are square matrices. A matrix is irreducible when it is not reducible.

Applying repeatedly the triangular decomposition to the resulting diagonal blocks, provided they are reducible, we obtain the Frobenius normal form of a reducible matrix [16]. In other words, we obtain the block triangular decomposition: where each diagonal block is either an irreducible square block or a null block of size .

The properties that we use in the study of the combined matrix of H-matrices are listed below (see [11]).

Theorem 2. Let be a nonsingular matrix and let be its combined matrix. Then(1)the sum by row and by column satisfies (2)if is a nonsingular diagonal matrix, (3)if and are permutation matrices, (4)if is a reducible matrix, then is a block diagonal matrix. More precisely, if is the Frobenius normal form of , then

Regarding the combined matrix of a reducible matrix, it is enough to study the combined matrix of each irreducible diagonal block.

Next we are going to recall basic definitions related to H-matrices.

Definition 3. The matrix is said to be diagonally dominant (DD) if and is called strictly diagonal dominant (SDD) if inequalities (10) are strict.

We are going to work with general H-matrices. Traditionally, H-matrices were considered only in the case where their comparison matrices are nonsingular M-matrices. Nevertheless, as we can see in [17], there are nonsingular H-matrices with singular comparison matrix. For example, is a nonsingular H-matrix, but its comparison matrix is a singular M-matrix. For this reason, we have to consider singular M-matrices in order to obtain all possible cases of nonsingular H-matrices.

The comparison matrix of a matrix is defined as therefore,

Definition 4. A real matrix is an M-matrix if and where represents the spectral radius of .

Definition 5. A matrix is called an H-matrix if it satisfies condition (14); that is, its comparison matrix is an M-matrix.

When the last inequality in (14) is strict, it follows that is a nonsingular M-matrix. Otherwise, is singular.

Analyzing the properties of nonsingular and singular H-matrices, a partition of the set of H-matrices in three classes is given in [17]:(1) belongs to the invertible class if and only if is a nonsingular M-matrix.(2) belongs to the mixed class if and only if is a singular M-matrix and its diagonal entries are not null.(3) belongs to the singular class if and only if is an M-matrix with at least one null diagonal entry.

The conclusions related to the singularity of each class are the following: all H-matrices in are nonsingular, in there are nonsingular and singular H-matrices, and all H-matrices in are singular and reducible. Moreover, if is irreducible, then all singular H-matrices in are diagonally equivalent to the singular matrix [18].

We study the combined matrix of nonsingular H-matrices. More precisely, we study the combined matrix of nonsingular H-matrices of both the invertible and the mixed classes.

We now recall the generalized diagonal dominance definitions.

Definition 6. A matrix is said to be generalized diagonally dominant (GDD) if there exists a nonnegative diagonal and nonsingular matrix of size such that is diagonally dominant; that is, and is called generalized strictly diagonal dominant (GSDD) if the inequalities in (15) are strict.

We have the following well known results (see [17, 19]).

Theorem 7. Let . Then,(1) if and only if it is GSDD;(2)if is GDD, then it is an H-matrix;(3)if and is irreducible, then it is GDD.

3. Combined Matrices of H-Matrices

In this section, we are going to extend to nonsingular and real H-matrices the following theorem that we have already commented on in the previous section (see [1]). All matrices considered in this section are real.

Theorem 8. The combined matrix of a nonsingular M-matrix is also a nonsingular M-matrix.

In fact, we are going to prove the diagonal dominance of the combined matrix of a nonsingular H-matrix.

3.1. Combined Matrices of H-Matrices of the Invertible Class

Let us start with H-matrices of the invertible class.

Theorem 9. Let . Then its combined matrix is strictly diagonal dominant.

Proof. Since , without loss of generality, we can suppose that is SDD; therefore, In addition, according to Theorem  2.5.12 in [11], we know that satisfies the strict inequalities We should notice that, with this same notation, the entries of the inverse matrix are equal to and inequality (17) can be expressed in the form Then, taking into account inequalities (16) and (19), it results that and consequently where are the entries of (3). This implies that the combined matrix is SDD.

Corollary 10. If , then .

Proof. It is straightforward.

It is well known that the class has different subclasses: - and -matrices, S-SDD, and so forth (see [20, 21]). Since all strictly diagonally dominant matrices belong to these subclasses, we can establish the following general corollary.

Corollary 11. Let be a subclass of H-matrices of the invertible class that contains all SDD matrices; that is, If , then belongs to the same subclass .

Proof. It is obvious.

It is worth noticing that this result can be obtained combining Theorems  3.1 and 3.5 of [12], where the authors work with the W-class of matrices satisfying condition (17).

Note that we can prove Theorem 8 as a consequence of Theorem 9.

Corollary 12. If is a nonsingular M-matrix, then is a nonsingular M-matrix.

Proof. Let us recall first that ; then is SDD. Since , has the signs pattern of ; consequently is a nonsingular M-matrix.

3.2. Combined Matrices of Nonsingular H-Matrices of the Mixed Class

We extend the previous results to nonsingular H-matrices of the mixed class. For this purpose, we need to extend two results of [11] on SDD matrices to nonsingular DD matrices.

We will denote by the sign of the real number .

Lemma 13. If is diagonally dominant and nonsingular, then

Proof. Since is DD and nonsingular, then for all .
Let us suppose first that for all . Given we build the matrix where .
Since is SDD because it is the sum of a DD matrix and a SDD matrix and , then (see [11, page 125]) and, therefore, . Since the determinant is a multilinear function, it is continuous, and since , then . Therefore, since is nonsingular.
Now let us suppose that there are some negative diagonal entries. We build the diagonal matrix Then, matrix has its diagonal entries positive and is DD. Therefore, applying the first part of this proof to we have .
Since , we conclude that

We should notice that if is singular and DD, then the sign of its determinant may not match with the sign of the diagonal entries product. This means that it can occur that , but . Besides, if , then would have at least one null row.

Lemma 13 can be extended to GDD matrices.

Lemma 14. If matrix is generalized diagonally dominant and nonsingular, then

Proof. It is enough to observe that if is a diagonal, nonsingular, and nonnegative matrix that transforms in a diagonally dominant matrix, then and the diagonal entries of are equal to .

The following result is essential for achieving our goals.

Theorem 15. Suppose that is diagonally dominant with nonzero diagonal entries. Then,

Proof. Let us start supposing that for all . In order to prove inequalities (29), let us consider, without loss of generality, that and . We are going to prove that . Then Let us prove that this auxiliary matrix is DD. In its first row, since is DD, we know that in particular, ; therefore, For the remaining rows, , it is true that Since is DD and, besides, all its diagonal entries are nonnegative, we can apply Lemma 13 and conclude that . Consequently ; then .
Finally, in the general case, when there are different signs in the diagonal entries, we build the sign matrix such that the matrix is DD and its diagonal entries are all positive. Applying the result of the first part of the proof to the matrix we conclude that for all . Finally, since we conclude that , , .

An immediate consequence of this result is the extension of Theorem  2.5.12 of [11] to DD matrices.

Theorem 16. Let be a nonsingular and diagonally dominant matrix. Then is diagonally dominant of its column entries. That is,

Proof. We know that a nonsingular and DD matrix does not have null diagonal entries (see [17]). Then, we can apply Theorem 15 to matrix and then inequalities (29) become inequalities (35).

Theorem 15 cannot be extended to GDD and to GSDD matrices because but . The following example illustrates this fact.

Example 17. The matrix is GSDD, because when it is multiplied by the diagonal matrix we have the following SDD matrix: Though the inverse matrix of satisfies inequality (35), the matrix does not satisfy it since and ; and . That is, and .

We are almost in position to extend the result on the combined matrix to nonsingular H-matrices of the mixed class.

Theorem 18. Let be a nonsingular and irreducible matrix. Then is diagonally dominant.

Proof. Since , we have to prove Since is an irreducible H-matrix of the mixed class, we know that is GDD by Theorem 7. As , we can suppose that is DD; that is, According to Theorem 15, Taking into account this last inequality in (39) we obtain (38) and consequently the combined matrix is DD.

Corollary 19. If is a nonsingular H-matrix, then its combined matrix is an H-matrix.

Proof. Suppose that is written in Frobenius normal form as it is shown in (5). Let us work with its diagonal blocks. If a diagonal block of belongs to , then its corresponding block in (see expression (9)) is SDD by Theorem 9 and belongs to by Corollary 10. Further, if an irreducible diagonal block of is in , then its corresponding block in is DD by Theorem 18. Then is an H-matrix.

Example 20. Let us consider the reducible and nonsingular H-matrix of the mixed class written in normal form, where the first diagonal block belongs to and the second diagonal block belongs to .
The combined matrix is Notice that is block diagonal. Each diagonal block is the combined matrix of the corresponding diagonal block of . Specifically, both blocks are SDD and then .

In conclusion, in this work, we have proved that the combined matrix of a nonsingular H-matrix of the class either or is an H-matrix. Actually, this is an extension of a well known result for nonsingular M-matrices.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referee for suggesting changes that have improved the presentation of the paper. This research was supported by Spanish DGI Grant no. MTM2014-58159-P.