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On the Connection between Kronecker and Hadamard Convolution Products of Matrices and Some Applications
Journal of Inequalities and Applications volume 2009, Article number: 736243 (2009)
We are concerned with Kronecker and Hadamard convolution products and present some important connections between these two products. Further we establish some attractive inequalities for Hadamard convolution product. It is also proved that the results can be extended to the finite number of matrices, and some basic properties of matrix convolution products are also derived.
There has been renewed interest in the Convolution Product of matrix functions that is very useful in some applications; see for example [1–6]. The importance of this product stems from the fact that it arises naturally in divers areas of mathematics. In fact, the convolution product plays very important role in system theory, control theory, stability theory, and, other fields of pure and applied mathematics. Further the technique has been successfully applied in various fields of matrix algebra such as, in matrix equations, matrix differential equations, matrix inequalities, and many other subjects; for details see [1, 7, 8]. For example, in , Nikolaos established some inequalities involving convolution product of matrices and presented a new method to obtain closed form solutions of transition probabilities and dependability measures and then solved the renewal matrix equation by using the convolution product of matrices. In , Sumita established the matrix Laguerre transform to calculate matrix convolutions and evaluated a matrix renewal function, similarly, in , Boshnakov showed that the entries of the autocovariances matrix function can be expressed in terms of the Kronecker convolution product. Recently in , Kiliçman and Al Zhour presented the iterative solution of such coupled matrix equations based on the Kronecker convolution structures.
In this paper, we consider Kronecker and Hadamard convolution products for matrices and define the so-called Dirac identity matrix which behaves like a group identity element under the convolution matrix operation. Further, we present some results which includes matrix equalities as well as inequalities related to these products and give attractive application to the inequalities that involves Hadamard convolution product. Some special cases of this application are also considered. First of all, we need the following notations. The notation is the set of all absolutely integrable matrices for all , and if , we write instead of . The notation is the transpose of matrix function . The notations and are the Dirac delta function and Dirac identity matrix, respectively; here, the notation is the scalar identity matrix of order . The notations , , and are convolution product, Kronecker convolution product and Hadamard convolution product of matrix functions and , respectively.
2. Matrix Convolution Products and Some Properties
In this section, we introduce Kronecker and Hadamard convolution products of matrices, obtain some new results, and establish connections between these products that will be useful in some applications.
Let , , and . The convolution, Kronecker convolution and Hadamard convolution products are matrix functions defined for as follows (whenever the integral is defined).
(ii)Kronecker convolution product
(iii)Hadamard convolution product
where is the th submatrix of order ; thus is of order , is of order , and similarly, the product is of order .
The following two theorems are easily proved by using the definition of the convolution product and Kronecker product of matrices, respectively.
Let , , , and let . Then for scalars and
Let , and let . Then
The above results can easily be extended to the finite number of matrices as in the following corollary.
Let and be matrices. Then
(i) The proof is a consequence of Theorem 2.3(v). Now we can proceed by induction on . Assume that Corollary 2.4 holds for products of matrices. Then
Similarly we can prove (ii).
Let , and let . Then
Here, and of order , is the th column of Dirac identity matrix with property . In particular, if , then we have
This completes the proof of Theorem 2.5.
Let . Then there exist two matrices of order and of order such that
is of order , is an matrix with all entries equal to zero, is an matrix of zeros except for a in the th position, and there are zero matrices between and (). In particular, if , then we have
The proof is by induction on . If , then the result is true by using (2.17). Now suppose that corollary holds for the Hadamard convolution product of matrices. Then we have
which is based on the fact that
and thus the inductive step is completed.
Let and be a matrix of zeros and that satisfies the (2.17). Then and is a diagonal matrix of zeros, and then the following inequality satisfied
It follows immediately by the definition of matrix .
Let and . Then for any matrix ,
By Corollary 2.7, it is clear that and so
This completes the proof of Theorem 2.8.
We note that Hadamard convolution product differs from the convolution product of matrices in many ways. One important difference is the commutativity of Hadamard convolution multiplication
Similarly, the diagonal matrix function can be formed by using Hadamard convolution multiplication with Dirac identity matrix. For example, if , and Dirac identity then we have
(i) if and only if and are both diagonal matrices;
3. Some New Applications
Now based on inequality (2.26) in the previous section we can easily make some different inequalities on using the commutativity of Hadamard convolution product. Thus we have the following theorem.
For matrices and and for , we have
In particular, if , then we have
Choose , where , and and are real scalars not both zero. Since
on using Theorem 2.5 we can easily obtain that
Now one can also easily show that
By setting , then it follows that ; further the arithmetic-geometric mean inequality ensures that and the choices and thus takes all values in . Now by using (3.4), (3.5) and inequality (2.26) we can establish Theorem 3.1.
Further, Theorem 3.1 can be extended to the case of Hadamard convolution products which involves finite number of matrices as follows.
Let . Then for real scalars , which are not all zero
where and with .
By taking indices "" and using (2.20) of Corollary 2.6 follows that
Now on using Corollary 2.6 and the commutativity of Hadamard convolution product yields
where and with then
Thus it follows that
Now by applying inequality (2.26), and (3.6) and (3.7) thus we establish Theorem 3.2.
We note that many special cases can be derived from Theorem 3.2. For example, in order to see that inequality (3.6) is an extension of inequality (3.2) we set and . Next, we recover inequality (3.1) of Theorem 3.1, by letting , then with , that is, then we have
By simplification we have
for every , just as required. Finally, if we let , , and , then on using Theorem 3.2 we have an attractive inequality as follows.
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The authors gratefully acknowledge that this research partially supported by Ministry of Science, Technology and Innovations(MOSTI), Malaysia under the Grant IRPA project, no: 09-02-04-0898-EA001. The authors also would like to express their sincere thanks to the referees for their very constructive comments and suggestions.
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Kılıçman, A., Al Zhour, Z. On the Connection between Kronecker and Hadamard Convolution Products of Matrices and Some Applications. J Inequal Appl 2009, 736243 (2009). https://doi.org/10.1155/2009/736243
- Finite Number
- Matrix Equation
- Matrix Function
- Matrix Algebra
- Kronecker Product