Advances in Matrices, Finite and Infinite, with Applications 2014View this Special Issue
Nonnegative Combined Matrices
The combined matrix of a nonsingular real matrix is the Hadamard (entrywise) product . It is well known that row (column) sums of combined matrices are constant and equal to one. Recently, some results on combined matrices of different classes of matrices have been done. In this work, we study some classes of matrices such that their combined matrices are nonnegative and obtain the relation with the sign pattern of . In this case the combined matrix is doubly stochastic.
Fiedler and Markham  studied matrices of the form , that is, the combined matrix of , where is a nonsingular matrix and means the Hadamard product. Combined matrices appear in the chemical literature where they represent the relative gain array (see ). Furthermore, the combined matrix gives the relation between the eigenvalues and diagonal entries of a diagonalizable matrix (see ). Results for the combined matrix of a nonsingular matrix and also for the Hadamard product have been obtained, for instance, in  where the behavior of the diagonal entries of the combined matrix of a nonsingular matrix was completely described and in  for the positive definite case.
It is well known  that the row and column sums of a combined matrix are always equal to one. Then, if is a nonnegative matrix, it has interesting properties and applications since it is a doubly stochastic matrix. For instance, in , there are two applications: the first one concerning a topic in communication theory called satellite-switched and the second concerning a recent notion of doubly stochastic automorphism of a graph. Recently, in , some implications on nonnegative matrices, doubly stochastic matrices, and graph theory, namely, graph spectra and graph energy, are presented.
Here, we focus our work on studying which matrices have nonnegative combined matrices. More precisely, we study the combined matrix of different classes of matrices such as totally positive and totally negative matrices and also when is totally nonnegative and totally nonpositive.
2. Notation and Previous Results
Unless otherwise indicated, in this work all square matrices are nonsingular and real. Given an matrix , we denote by the subset of indexes . For the subsets , the submatrix with rows lying in the subset and columns in the subset is denoted by , and the principal submatrix is abbreviated to . Similarly, denotes the submatrix obtained from by deleting rows lying in and columns in , and is abbreviated to . Then, denotes the submatrix obtained from by deleting row and column , and . Moreover, denotes the minor; that is, .
To avoid confusion with other matrices we will say that is a nonnegative (positive) matrix if it is entrywise nonnegative (positive); that is, () for all , and we will denote it by ). Similar notation will be used for the nonpositive (negative) case; that is, () for all , and we will denote it by ().
Now we recall some classes of matrices that we are working on.
Definition 1. An real matrix is said to have a checkerboard pattern if or for all . When no entry is zero, one will say that is strictly checkerboard.
Definition 2. A matrix is said to be totally positive (negative) if all its minors of any order are positive (negative). That is, for every subset , (). It is denoted by TP (TN).
Definition 3. A matrix is called totally nonpositive (nonnegative) if all its minors of any order are nonpositive (nonnegative). That is, for every subset , (). It is denoted by TNP (TNN).
Definition 4. An real matrix is said to be a -matrix if, for every subset , .
Definition 5. An real matrix is called an -matrix if can be written as , with and , where denotes the spectral radius of matrix , that is, the biggest absolute value of the eigenvalues of .
For a matrix , denotes its comparison matrix; that is,
Definition 6. An complex matrix is called an -matrix if its comparison matrix is an -matrix.
Note that nonsingular -matrices having singular comparison matrix are included in this definition (see ).
Remember that the Hadamard (or entrywise) product of two matrices and is the matrix .
Definition 7. The combined matrix of a nonsingular real matrix is defined as . Then, if , The elements of will be denoted by .
It is clear that the combined matrix has the following properties: (i) and (ii) is doubly stochastic if is nonnegative.
3. Matrices with
Looking at the definition of a combined matrix it is easy to see that the combined matrix of a triangular matrix is the identity matrix. Moreover, the following result from Lemma (page ) of  can be established.
Lemma 8. For any nonsingular matrix , for any two permutation matrices and , and for any triangular matrix one has(i),(ii); that is, is a permutation matrix,(iii).
From Definition 7 one can observe that, for and , if and only if the matrix is permutationally similar to a triangular matrix. The natural question is whether or not this is a general equivalence. As it is suggested in  (Problem 2, page 302) the equivalence is not true for some matrices with as the following example shows.
Example 9. Consider the matrix It is easy to compute and to see that . However, is not permutationally similar to a triangular matrix since is irreducible.
Proposition 10. Let be a nonsingular matrix. If the combined matrix is nonnegative and triangular, then .
Proof. Recall that is doubly stochastic since it is nonnegative. Suppose is upper triangular. In this case is the only positive entry of the first column and the first row. Reasoning as before with the other columns of we conclude that .
A simple case to determine matrices having their combined matrix is given for matrices in the following result.
Proposition 11. Let be a nonsingular matrix. Then In particular, (i) if , then , (ii) if , then , and finally (iii) if , then .
() If , then and . Therefore, .
() Suppose first that . If , since , then , in which case . Then and and hence . Similarly, if , we obtain and and then .
The two first particular cases come from the fact that is doubly stochastic, and the last case is because all entries are different from zero.
The positivity of the matrix may play an important role in our case. According to the definition of the combined matrix, we have the following.
Theorem 13. Let (). Then if and only if ().
Matrices with the property described in Theorem 13 are necessarily nonnegative monomial matrices, as are the corresponding combined matrices. Monomial matrices are indeed permutationally similar to diagonal matrices and thus are orthogonal matrices. Below we prove a related result about a type of orthogonal matrices that was recently introduced in .
Definition 14. A nonsingular matrix is called a -matrix if two nonsingular diagonal matrices and exist such that .
Theorem 15. Let be a nonsingular -matrix such that and . Then, the combined matrix of is nonnegative.
Proof. In this case, we have which is nonnegative.
Example 16. The matrix is a -matrix, since is the result of , where . Then, the combined matrix is
It is well known that the combined matrix of an -matrix is an -matrix (see ). The following result gives the equivalent conditions to have a nonnegative combined matrix of an -matrix.
Theorem 17. Let A be an -matrix. Then, the following conditions are equivalent:(i),(ii) is diagonal,(iii),(iv) for all .
Proof. Since is -matrix, then is nonnegative and so the off-diagonal elements of are nonpositive and the diagonal elements are positive. Then, if and only if is diagonal. The last two equivalences follow because the combined matrix is doubly stochastic.
Despite the fact that the nonnegativity of the combined matrix of an -matrix is reduced to the identity matrix, it is easy to find -matrices for which its combined matrix is nonnegative and different from the identity matrix as the following example shows.
Example 18. The nonsingular matrices are -matrices and each one has positive combined matrix: It should be noted that is nonsingular but is a singular -matrix.
Now, we study the positivity of the combined matrix of totally positive and totally negative matrices.
Theorem 19. If is a TP-matrix, then is strictly checkerboard. In addition, if is a TN-matrix, then is strictly checkerboard.
Proof. It is straightforward noting that is checkerboard in both cases.
Then, the combined matrix of a TP-matrix or a TN-matrix is not nonnegative. However, we have the following result.
Theorem 20. If is a TP-matrix (TN-matrix), then (), where .
To study the combined matrix of a TNN-matrix, we need some auxiliary results.
Proposition 21 (see , Corollary 3.8). If is a nonsingular TNN-matrix, then is a P-matrix.
Proposition 22. If is a nonsingular TNN-matrix and for some , then .
Proof. If , then , which is positive since is a -matrix by Proposition 21. For , consider the submatrix and let us work by contradiction. Suppose that . Since is a nonsingular TNN-matrix, then and . Then which is a contradiction.
Proposition 23. If is a nonsingular TNN-matrix, then the following conditions are equivalent:(i),(ii), , .
Proof. Since is TNN, and for all . Then Therefore, is nonnegative if and only if , when .
Proposition 24 (see , Theorem 3.3). If is a nonsingular TNN-matrix, then is also a nonsingular TNN-matrix, where .
Then we can establish the following result.
Theorem 25. Let be a nonsingular TNN-matrix. Then the following conditions are equivalent:(i) is nonnegative,(ii).
Proof. . If , it is obvious that is nonnegative.
. Let us suppose that is nonnegative. We note that whenever , , by Proposition 23. Then it remains to prove that whenever and . We work by contradiction. For this, we suppose that there exists with and . By Proposition 24 is a nonsingular TNN-matrix. Applying Proposition 22 to both and we have that and . Therefore, , with , and this contradicts the result of Proposition 23. Thus, when , we have that . Then, is a lower triangular and nonnegative matrix. Hence, by Proposition 10.
Now, let us figure out the combined matrix of a TNP-matrix. Since all minors of TNP-matrices are nonpositive, it is clear that the combined matrix of a nonsingular TNP-matrix is not, in general, nonnegative. In fact, we have this simple result.
Theorem 26. Let be a nonsingular TNP-matrix. Then is nonnegative if and only if , whenever , .
However, there exist TNP-matrices with nonnegative combined matrices as the following example shows.
Example 27. Consider the totally nonpositive matrix The combined matrix is which is nonnegative.
The question now is to know whether or not there are more TNP-matrices with nonnegative combined matrices. As we see below, only TNP-matrices of size may have this property. To prove this we need some auxiliary results concerning TNP-matrices.
Proposition 28. Let be an nonsingular TNP-matrix with . If there exists an index such that , then .
Proof. Suppose by contradiction that there is an and for some . Since is nonsingular, there exists an index such that and then which is a contradiction with the signature of the matrix (see ).
Corollary 29. Let be an nonsingular TNP-matrix with .(i)If for some , then for all .(ii)If for some , then , .
Proof. (i) The proof follows from Proposition 28.
(ii) Since is TNP, for all (see , Theorem 2.1 (i)). Then, using part (i) of this corollary, we conclude that whenever .
In the following theorem we show how the first row and the second column of a nonnegative combined matrix of a TNP-matrix are.
Theorem 30. Let be an nonsingular TNP-matrix with . If is nonnegative, then the first row of is and the second column is .
Proof. Again, since is TNP, for all . Furthermore, , , by Theorem 26 and for by Proposition 28. Then, the proof follows since is doubly stochastic and .
Now, we can give the main theorem on combined matrices of TNP-matrices; that is, we are going to prove that there does not exist any nonsingular TNP-matrix of size such that its combined matrix is nonnegative.
Theorem 31. Let be an nonsingular TNP-matrix with . Then is not nonnegative.
Proof. Let us work by contradiction and suppose that there exists a nonsingular TNP-matrix of size , with and . By Theorem 30 we know the structure of the first row and the second column of . Let us focus on its first column where its first entry is . Since is nonnegative and so doubly stochastic, then there must exist an index such that , and then . Further, by Theorem 30 and Theorem 2.1 (i) of  and by Theorem 30. Then This is a contradiction with signature of according to .
Finally, we prove what kind of totally nonpositive matrices of size have their combined matrix nonnegative in the following result.
Theorem 32. Let the TNP-matrix be where , , , are nonnegative. Then if and only if at least one of the entries or is zero. In this case, .
Proof. The computation of the combined matrix gives Then, since , if and only if . In this case, and .
In brief, only antitriangular TNP-matrices have their combined matrix nonnegative.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for their suggestions that have improved the reading of this paper. This research is supported by Spanish DGI (Grant no. MTM2010-18674).
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