On the Connection between Kronecker and Hadamard Convolution Products of Matrices and Some Applications

Abstract

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.

1. Introduction

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 [2], 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 [6], Sumita established the matrix Laguerre transform to calculate matrix convolutions and evaluated a matrix renewal function, similarly, in [9], Boshnakov showed that the entries of the autocovariances matrix function can be expressed in terms of the Kronecker convolution product. Recently in [1], 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.

Definition 2.1.

Let , , and . The convolution, Kronecker convolution and Hadamard convolution products are matrix functions defined for as follows (whenever the integral is defined).

(i)Convolution product

(2.1)

(ii)Kronecker convolution product

(2.2)

(iii)Hadamard convolution product

(2.3)

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.

Theorem 2.2.

Let , , , and let . Then for scalars and

• (i)
  

  (2.4)

• (ii)
  

  (2.5)

• (iii)
  

  (2.6)

• (iv)
  

  (2.7)

Theorem 2.3.

Let , and let . Then

• (i)
  

  (2.8)

• (ii)
  

  (2.9)

• (iii)
  

  (2.10)

• (iv)
  

  (2.11)

• (v)
  

  (2.12)

• (vi)
  

  (2.13)

The above results can easily be extended to the finite number of matrices as in the following corollary.

Corollary 2.4.

Let and be matrices. Then

• (i)
  

  (2.14)

• (ii)
  

  (2.15)

Proof.

(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

(2.16)

Similarly we can prove (ii).

Theorem 2.5.

Let , and let . Then

(2.17)

Here, and of order , is the th column of Dirac identity matrix with property . In particular, if , then we have

(2.18)

Proof.

Compute

(2.19)

This completes the proof of Theorem 2.5.

Corollary 2.6.

Let . Then there exist two matrices of order and of order such that

(2.20)

where

(2.21)

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

(2.22)

Proof.

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

(2.23)

which is based on the fact that

(2.24)

and thus the inductive step is completed.

Corollary 2.7.

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

(2.25)

Proof.

It follows immediately by the definition of matrix .

Theorem 2.8.

Let and . Then for any matrix ,

(2.26)

Proof.

By Corollary 2.7, it is clear that and so

(2.27)

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

(2.28)

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;

(ii).

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.

Theorem 3.1.

For matrices and and for , we have

(31)

In particular, if , then we have

(32)

Proof.

Choose , where , and and are real scalars not both zero. Since

(33)

on using Theorem 2.5 we can easily obtain that

(34)

Now one can also easily show that

(35)

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.

Theorem 3.2.

Let . Then for real scalars , which are not all zero

(36)

where and with .

Proof.

Let

(37)

By taking indices "" and using (2.20) of Corollary 2.6 follows that

(38)

Now on using Corollary 2.6 and the commutativity of Hadamard convolution product yields

(39)

where and with then

(3.10)

Thus it follows that

(3.11)

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

(3.12)

By simplification we have

(3.13)

for every , just as required. Finally, if we let , , and , then on using Theorem 3.2 we have an attractive inequality as follows.

(3.14)