Combined matrix of diagonally equipotent matrices

: Let A A A T (cid:2) ( ) = ∘ − be the combined matrix of an invertible matrix A , where ∘ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an H -matrix whose comparison matrix is singular. In particular, we focus on A (cid:2) ( ) when A is diagonally equipotent, and we study whether A (cid:2) ( ) is an H -matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices


Introduction
Let A be an n n × nonsingular matrix, and let A ( ) be its combined matrix (see [8]), that is, where ∘ means the Hadamard product of matrices.The name of combined matrix appears for the first time in [8].In control theory, it is known as "relative gain array" [4].Moreover, the combined matrix has been applied in other subjects: (i) for constructing doubly diagonally stochastic matrices [10], (ii) to give a relationship between the eigenvalues and the diagonal entries [13], and (iii) to build a lower bound of the condition number [7].
Recall that the combined matrix of a reducible invertible matrix A is a block diagonal matrix whose diagonal blocks are the combined matrix of each irreducible diagonal block of the normal form of A (see [6]).Then, in this article, we always work with irreducible nonsingular matrices.We recall three basic properties of combined matrices: The entry sum of each row and column is 1.
Various concepts of diagonal dominance of a matrix are essential in this work and are reviewed next.
∈ × is said to be (i) generalized diagonally dominant (GDD) if there exists an invertible nonnegative diagonal matrix D of size n such that AD is diagonally dominant (DD); that is, (ii) generalized strictly diagonally dominant (GSDD) if all the aforementioned inequalities are strict.
(iii) generalized diagonally equipotent (GDE) if all the aforementioned inequalities are equalities.
Note that when the diagonal invertible matrix D is the identity matrix, the aforementioned three definitions become DD, strictly diagonally dominant (SDD), and diagonally equipotent (DE), respectively.Any DE matrix is denoted by DE as in [2].In addition, we say that A is irreducible DD if the matrix is irreducible and at least one of the aforementioned inequalities is strict [17,Definition 1.7].The rows have been considered in all those definitions.
Related to diagonal properties is the set of H -matrices (Section 2).We consider the three classes of H -matrices defined in [5].The GSDD property is useful in the study of H -matrices of the invertible class ( I ), that is, H -matrices whose comparison matrix is an invertible M -matrix.
We also work with H -matrices in the mixed class ( M ) whose comparison matrix is a singular M -matrix, and there exists an invertible equimodular matrix.In both cases, we consider that all matrices are irreducible and so with nonnull diagonal elements since the class of a reducible H -matrix is determined by the properties of each square diagonal block [5, Theorems 3 and 5].The class of singular H -matrices such that all equimodular matrices are singular is not considered.
In [6], the diagonally dominance of the combined matrix of H -matrices is studied.When the matrix is in the invertible class ( I ), it follows that the combined matrix is an H -matrix in the same class I [6, Corollary 10] as it was given in [13, Lemma 5] and the combined matrix of an invertible matrix in M is an H -matrix [6,Corollary 19].This last result does not differentiate between being in I or in M for the corresponding combined matrix.
In this work, we study the combined matrix of an invertible H -matrix in the mixed class ( M ) shedding some light on the possible mentioned difference.In Section 2, we present some results on DE matrices.In Section 3, we study the combined matrix in some particular cases of these kinds of H -matrices denoted by DpM.In Section 4, we go one step further studying the combined of matrices in M denoted by DpMp1, that is, DpM matrices that have one more nonzero entry.In Section 5, we give our conclusions.

DE matrices
Let us see first that any irreducible and nonsingular H -matrix in M is DE.Recall that the comparison matrix, of an . First, we recall in Theorem 1 the relationship between the diagonally dominance property and general H -matrices.The first proposition can be seen in [5,Theorem 4].The second part can be seen, for instance, in [6,14], where the results of the basic paper [16] were used.Theorem 1.Let A be an irreducible matrix.Then (i) A is GDD if and only if A is an H-matrix.(ii) A is GSDD if and only if A is in I .Now, we have the following result for irreducible H -matrices in M .Theorem 2. Let A be an irreducible matrix.The following conditions are equivalent , its comparison matrix A ( ) is a singular M -matrix.Then there exists a positive vector d such that A d 0 ( ) = by [3,Theorem 6.4.16].This property is essentially , the aforementioned equation shows that the matrix AD is DE, and hence A is GDE.(ii) ⇒ (i).The matrix A is an H -matrix since it is GDD by part (i) of Theorem 1.Moreover, the generalized diagonal dominance is not strict in any, row then A is not an I by part (ii) of Theorem 1.Then, the irreducible matrix A is in M .□ It is worth saying that in the set of irreducible H -matrices, we have the following characterizations: Theorem 3. Let A be an irreducible matrix, of order n, in M .Let A ( ) be its comparison matrix.Then, the cofactors M ij of A ( ) are constant in each row, that is, … , .
ii ij

= =
Proof.Since A is in M then A ( ) is singular.Then, by using the Lagrange expansion along the ith row, with the same entries as B ( ), ii ik = , even if a 0 ik = , for the given index i.The aforementioned reasoning can be done for all rows, that is, for all i n 1, 2,…, = .Hence, the cofactors of B ( ) are constant in each row.□

Combined of DpM matrices
Let us start with the definition of DpM matrices.For that, we will work with monomial matrices.A square matrix A is monomial if A DP = , where P is a permutation matrix and D is an invertible diagonal matrix.
Combined matrix of diagonally equipotent matrices  3 Definition 2. Let A be a matrix of order n.We say that A is DpM if can be written as the sum of a nonsingular Diagonal matrix plus an irreducible Monomial matrix, that is, where D is a nonsingular diagonal matrix and M is an irreducible monomial matrix.
In the following steps, we prepare our initial matrix A using the two basic properties of combined matrices since we are going to study the combined matrix of a nonsingular (irreducible) DpM matrix in M .
First step.We are working with matrix A in M which means that the matrix AD 1 is DE by Theorem 2, for some invertible nonnegative diagonal matrix D 1 .The combined matrix of both matrices are equal by property (1).Then, we are working with a DE matrix.
Second step.With a convenient permutation matrix P, our matrix A can be written as follows: by property (2) of combined matrices, we assume that our invertible matrix in M has the above structure.
Third step.Finally, all diagonal elements are nonzero since our nonsingular H -matrix is irreducible by [5, Theorem 3].We can multiply our matrix A on the right by the diagonal matrix ∕ and obtain the following matrix: where x 1 , since our matrix is DE.Note that by Property 1. Accordingly, we suppose that our matrix in the following results of this section is the matrix A of (5).Theorem 4. Let A be a nonsingular DpM matrix in M of order n.Then, there exists a permutation matrix P such that Proof.Suppose we have made all operations, of the last three steps, to our initial nonsingular irreducible DpM matrix in M to have the structure (5), included a symmetric permutation with the permutation matrix P.
The general expression of the determinant of a square matrix where n is the set of all permutations σ of the n first natural numbers and σ i denotes the position of the ith number after the permutation σ .The determinant of our matrix (5) is where the symbol "sgn" stands for the sign of the permutation n 2, 3, …, , 1 { } of the n first natural numbers.We have

= = ⋅
Reasoning recursively with these two properties, we obtain that all nonnull entries of the combined matrix are exactly 1 2 ⋅ Consequently, the combined matrix of the initial DpM matrix in M is a symmetric permutation of (6).□ Theorem 5. Let A be a nonsingular DpM matrix in M of order n.Then, A ( ) is an irreducible Proof.We assume that our matrix A is given by (5).From the general expression of the determinant of a square matrix M m given in (7) and observing the structure of the matrix (6), we note that there are only two permutations n n 1, 2, …, and 2, 3, …, , 1 , where the product }= , the determinant of the combined matrix A ( ) given in ( 6) is }= − .In addition, the combined matrix is irreducible and DE by (6).Hence, the irreducible matrix A ( ) is in M by Theorem 2. □ Note that the cofactors of the companion matrix of A ( ), that is, A ( ( )) are equal in each row by Theorem 3.For instance, if we suppose that the matrix (6) has order 5, then the cofactors of row one are equal to 0.0625.
Define the map A A Φ ( ) ( ) = . In [13], the limit is studied for H -matrices with nonsingular comparison matrix, that is, matrices in the invertible class.
In our case, that is, when we are working with nonsingular H -matrices in M , we have different results.
Combined matrix of diagonally equipotent matrices  5 Theorem 7. Let A be a nonsingular DpM matrix in M of order n and consider Proof.The combined matrix of A is a symmetric permutation of ( 6) by Theorem 4.
(i) If n k 2 1 = + , then A ( ) given by ( 6) is invertible by Theorem 4. It is clear that the invertible A ( ) is a DpM matrix in M and can be considered as (5) using the diagonal matrix D diag . By applying Theorem 5, we have , there is nothing to prove because A ( ) given in ( 6) is singular by Theorem 5. Therefore, the limit does not exist.□

Combined of DpMp1 matrices
Now, let us go one step further in the study of combined matrices of H -matrices in M .For that, we are going to work with DpM matrices when we add exactly one nonzero entry.

Definition 3.
Let A be a matrix of order n.We say that A is DpMp1 if is DpM with exactly one more nonzero entry in any place of the zero pattern of A.
Note that, a DpMp1 matrix can be written as the sum of a nonsingular Diagonal matrix plus an irreducible Monomial matrix plus 1 nonzero entry as follows: where D is a nonsingular diagonal matrix, M is an irreducible monomial matrix, and E ij is a matrix with exactly one nonzero entry in the i j , ( )th position, i j ≠ , not included in the nonzero pattern of M .The first thing we note in this case is that despite our matrix A being irreducible, its combined matrix can be reducible as seen in the following example.

Example 1. The DpMp1 matrix
is a triangular matrix and thereby reducible.
Again, some manipulations are needed in our DpMp1 invertible matrix in M to obtain our result.First step.Our initial matrix A is in M .It means that the matrix AD 1 is DE, for some nonnegative nonsingular diagonal matrix D 1 .Since the combined matrix of both matrices are equal, that is, A AD , we will work with a DE matrix in the proof.
Second step.With a convenient symmetric permutation our matrix can be written as follows: where we have denoted all nonzero entries by a ij for simplicity, and we have chosen the new nonzero entry in the upper right corner, without loss of generality.Third step.Finally, as all diagonal elements are nonnull since our matrix in M is irreducible, we can multiply our matrix by the diagonal matrix D a diag 1 ii 2 ( ) = ∕ and obtain the matrix For now on, we suppose our matrix A will have the three operations made in the three steps and will work with the matrix (10), with the conditions (11).Matrices DpMp1 in M can have the following determinants.
Theorem 8. Let A be a matrix DmMp1 in M of order n as the matrix (10).Then, given four possibilities: (1) A det 2 = , and (4) A det 0 = .
Proof.Again, from the general expression of the determinant of a matrix (7) and observing the structure of the matrix (10), we note that there are only three permutations

Let us see the four possible values of (12). (1) If
Combined matrix of diagonally equipotent matrices  7 (3) If − , we have a singular matrix since A det 0 = by (11).□ Remark 1. Theorem 8 remains valid if the new nonzero entry y will appear in any other position of the upper triangular part of A since the corresponding entries x i , x and y will satisfy conditions (11).Moreover, the same result will happen if the new nonzero entry is in the lower triangular part.
Theorem 9. Let A be an invertible matrix DpMp1 in M of order n.Then, (i) A ( ) is GDD with at least one row strictly diagonal dominant; (ii) the combined matrix A ( ) is an H-matrix in I .
Proof.By properties (1-2) of a combined matrix, we can assume that we are working with a matrix as (10).We consider three cases according to the nonzero values of A det given in Theorem 8. Case 1.
A x det 2| | = .Proof of (i).The entries of A ( ) are as follows: where A ij is the cofactor of a ij .Then, we have Note that the last row satisfies the inequality

and m 1
ik = − .The irreducible matrix B ( ) is a comparison matrix and so H -matrix, which is DE and then singular.Then, reasoning as before with this new matrix, we have M M , ii ik = since m 0 ik ≠ .In addition, the cofactors of the ith row of B ( ) are the same than those of B ( ).Then, M M since A 11 is the determinant of a triangular submatrix with 1's in the diagonal.By property (3) of a combined matrix, we have using the Laplace expansion of the determinant of the matrix(10), by the cofactors of the first row, we have By applying property (3) of combined matrices to the second column, we have c Working in this way, with the following rows and columns, we obtain [12]e the cofactor of the triangular block A 11 is 1.Recalling that the combined matrix preserves the zero pattern of A and the sum of the entries of each row and column is 1[12], we have