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Volume 2009, Article ID 736243, 10 pagesdoi:10.1155/2009/736243 Research Article On the Connection between Kronecker and Hadamard Convolution Products of Matrices and Some Applications 1

Volume 2009, Article ID 736243, 10 pages

doi:10.1155/2009/736243

Research Article

On the Connection between Kronecker and

Hadamard Convolution Products of Matrices

and Some Applications

1 Department of Mathematics, Institute for Mathematical Research, University Putra Malaysia,

43400 UPM Serdang, Selangor, Malaysia

2 Department of Mathematics, Zarqa Private University, P.O Box 2000, Zarqa 1311, Jordan

Correspondence should be addressed to Adem Kılıc¸man,akilicman@putra.upm.edu.my

Received 16 April 2009; Revised 29 June 2009; Accepted 14 July 2009

Recommended by Martin J Bohner

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

Copyrightq 2009 A Kılıc¸man and Z Al Zhour This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

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 see1,7,8 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, in9, Boshnakov showed that the entries of the autocovariances matrix function can be expressed

in terms of the Kronecker convolution product Recently in 1, Kilic¸man 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 D n t 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 M I

m,n is the set of all m × n absolutely integrable matrices for all

t ≥ 0, and if m  n, we write M I

n instead of M I

m,n The notation A T t is the transpose of matrix function At The notations δt and D n t  δtI nare the Dirac delta function and

Dirac identity matrix, respectively; here, the notation I nis the scalar identity matrix of order

n ×n The notations At∗Bt, AtBt, and At•Bt are convolution product, Kronecker convolution product and Hadamard convolution product of matrix functions At and Bt,

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 At  f ij t ∈ M I

m,n , Bt  g jr t ∈ M I

n,p , and Ct  z ij t ∈ M I

The convolution, Kronecker convolution and Hadamard convolution products are matrix

functions defined for t≥ 0 as follows whenever the integral is defined

i Convolution product

At ∗ Bt  h ir t with h ir t n

k1

t

0

f ik t − xg kr xdx n

k1

f ik t ∗ g kr t. 2.1

ii Kronecker convolution product

At  Bt f ij t ∗ Btij 2.2

iii Hadamard convolution product

At•Ct f ij t ∗ z ij tij 2.3

where f ij t ∗ Bt is the ijth submatrix of order n × p; thus At  Bt is of order mn × np, At ∗ Bt is of order m × p, and similarly, the product At•Ct is of order m × n.

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 At, Bt, Ct ∈ M I

n , and let D n t  δtI n ∈ M I

n Then for scalars α and β

i



αAt  βBt∗ Ct  αAt ∗ Ct  βBt ∗ Ct, 2.4

ii

At ∗ Bt ∗ Ct  At ∗ Bt ∗ Ct, 2.5

iii

At ∗ D n t  D n t ∗ At  At, 2.6

iv

At ∗ Bt T  B T t ∗ A T t. 2.7

Theorem 2.3 Let At, Ct ∈ M I

m,n , Bt ∈ M I

p,q , and let D n t  δtI n ∈ M I

n Then

i

D n t  At  diagAt, At, , At, 2.8

ii

D n t  D m t  D nm t, 2.9

iii

At  Ct  Bt  At  Bt  Ct  Bt, 2.10

iv

At  Bt T  A T t  B T t, 2.11

v

At  Bt ∗ Ct  Dt  At ∗ Ct  Bt ∗ Dt, 2.12

vi

At  D m t ∗ D n t  Bt  D n t  Bt ∗ At  D m t  At  Bt. 2.13

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

Corollary 2.4 Let A i t and B i t ∈ M I

n 1 ≤ i ≤ k be matrices Then

i

k



i1 ∗ A i t  B i t 

k

i1

∗ A i t 

k

i1

∗ B i t , 2.14

ii

k



i1  A i t ∗ B i t 

k



i1

 A i t ∗

k



i1

 B i t 2.15

Proof. i The proof is a consequence ofTheorem 2.3v Now we can proceed by induction

on k Assume thatCorollary 2.4holds for products of k− 1 matrices Then

A1t  B1t ∗ A2t  B2t ∗ · · · ∗ A k t  B k t

 {A1t  B1t ∗ A2t  B2t ∗ · · · ∗ A k−1t  B k−1t} ∗ A k t  B k t

 {A1t ∗ A2t ∗ · · · ∗ A k−1t  B1t ∗ B2t ∗ · · · ∗ B k−1t} ∗ A k t  B k t

 {A1t ∗ A2t ∗ · · · ∗ A k−1t ∗ A k t}  {B1t ∗ B2t ∗ · · · ∗ B k−1t ∗ B k t}



k



i1

∗ A i t 

k



i1

∗ B i t

2.16

Similarly we can proveii

Theorem 2.5 Let At  f ij t, and let Bt  g ij t ∈ M I

m,n Then

A •Bt  P T

m t ∗ A  Bt ∗ P n t. 2.17

Here, P n t  Vec E n11t, , Vec E n nn t ∈ M n2,n and E ij t  e i t ∗ e T

j t of order n × n, e i t

is the ith column of Dirac identity matrix D n t  δtI n ∈ M n with property P T t ∗ P n t  D n t.

In particular, if m  n, then we have

A •Bt  P T

n t ∗ A  Bt ∗ P n t. 2.18

Proof Compute

P m T t ∗ A  Bt ∗ P n t Vec E11m t, , Vec E m mm tT ∗ A  Bt

∗Vec E n11t, , Vec E n nn t

n

k1

diag

f ik t, f 2k t, , f mk t∗ Bt ∗ E n kk t



n



k1

f ik t ∗ g ij t ∗ δ jk t f ij t ∗ g ij t A•Bt.

2.19

This completes the proof ofTheorem 2.5

Corollary 2.6 Let A i t ∈ M I

m,n 1 ≤ i ≤ k, k ≥ 2 Then there exist two matrices P km t of order

m k × m and P kn t of order n k × n such that

k



i1

•A i t  P T

km t ∗

k

i1

 A i t ∗ P kn t, 2.20

where

P km T t E m11 t, 0 m , , 0 m , E22m t, 0 m , , 0 m , E m mm t 2.21

is of order m ×m k , 0 m is an m ×m matrix with all entries equal to zero, E m ij t is an m×m matrix of zeros except for a δt in the ijth position, and there are k−2

s1m s zero matrices 0 m between E ii m t and E m i 1,i1 t (1 ≤ i ≤ m − 1) In particular, if m  n, then we have

k



i1

•A i t  P T

km t ∗

k

i1

 A i t ∗ P km t. 2.22

Proof The proof is by induction on k If k  2, then the result is true by using 2.17 Now

suppose that corollary holds for the Hadamard convolution product of k matrices Then we

have

k1



i1

•A i t  A1t•

k1



i1

•A i t  P T

m t ∗

A1t 

k1



i1

•A i t ∗ P n t

 P T

m t ∗

D m t  P T

k1



i1

 A i t ∗ D n t  P kn t ∗ P n t

P m T t ∗D m t  P T

k1



i1

 A i t ∗ D n t  P kn t ∗ P n t,

2.23

which is based on the fact that

P T

m t ∗D m t  P T

k1m t, D n t  P kn t ∗ P n t  P k1n t, 2.24 and thus the inductive step is completed

Corollary 2.7 Let At, Bt ∈ M I

m and P m t be a matrix of zeros and D m t that satisfies the

2.17 Then P T

m t ∗ P m t  D m t and P m ∗ P T

m is a diagonal m2× m2matrix of zeros, and then the following inequality satisfied

0≤ P m t ∗ P T

m t ≤ D m2. 2.25

Proof It follows immediately by the definition of matrix P m t.

Theorem 2.8 Let At and Bt ∈ M I

m,n Then for any m2× n2matrix Lt,

P m T t ∗ Lt ∗ L T t ∗ P m t ≥P m T t ∗ Lt ∗ P n t∗P m T t ∗ Lt ∗ P n tT ≥ 0. 2.26

Proof ByCorollary 2.7, it is clear that D n2t ≥ P n t ∗ P T t ≥ 0 and so

P m T t ∗ Lt ∗ D n2t ∗ L T t ∗ P m t  P T

m t ∗ Lt ∗ L T t ∗ P m t

≥ P T

m t ∗ Lt ∗ P n t ∗ P T

n t ∗ L T t ∗ P m t

P m T t ∗ Lt ∗ P n t∗P m T t ∗ Lt ∗ P n tT ≥ 0.

2.27 This completes the proof ofTheorem 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

A •Bt  B•At. 2.28

Similarly, the diagonal matrix function can be formed by using Hadamard convolution

multiplication with Dirac identity matrix For example, if At, Bt ∈ M I

n , and D n t Dirac

identity then we have

i A•Bt  A ∗ Bt if and only if At and Bt are both diagonal matrices;

ii A•Bt•D n t  A•D n t ∗ B•D n t.

3 Some New Applications

Now based on inequality2.26 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 At and Bt ∈ M I

m,n and for s ∈ −1, 1, we have At ∗

A T t•Bt ∗ B T t  sAt ∗ B T t•Bt ∗ A T t

≥ 1  sAt•Bt ∗ At•Bt T

In particular, if s  0, then we have

At ∗ A T t•Bt ∗ B T t≥ At•Bt ∗ At•Bt T 3.2

Proof Choose Lt  αAt  Bt  βBt  At, where At, and Bt ∈ M I

m,n and α, β are real

scalars not both zero Since

Lt ∗ L T t 

αAt  Bt  βBt  At∗αAt  Bt  βBt  AtT

, 3.3

on usingTheorem 2.5we can easily obtain that

P m T t ∗ Lt ∗ L T t ∗ P m t α2

At ∗ A T t•Bt ∗ B T t

αβ

At ∗ B T t•Bt ∗ A T t

αβ

Bt ∗ A T t•At ∗ B T t

β2

Bt ∗ B T t•At ∗ A T t

α2 β2

At ∗ A T t•Bt ∗ B T t

 2αβ At ∗ B T t•Bt ∗ A T t .

3.4

Now one can also easily show that

P m T t ∗ Lt ∗ P n t∗P m T t ∗ Lt ∗ P n tTα  β2At•Bt ∗ At•Bt T 3.5

By setting s  2αβ/α2β2, then it follows that s1  α  β2/α2β2; further the arithmetic-geometric mean inequality ensures that|s| ≤ 1 and the choices β  1 and α ∈ −1, 1 thus s

takes all values in−1, 1 Now by using 3.4, 3.5 and inequality 2.26 we can establish

Theorem 3.1

Further,Theorem 3.1can be extended to the case of Hadamard convolution products which involves finite number of matrices as follows

Theorem 3.2 Let A i ∈ M I

m,n 1 ≤ i ≤ k, k ≥ 2 Then for real scalars α1, α2, , α k , which are not all zero

k



i1

α2i

k



i1

•A i t ∗ A T

k−1



r1

μ r k



•A w t ∗ A T

wrt

≥

k

i1

α i

2 k

i1

•A i t k

i1

•A i t

T

,

3.6

where μ r  k

w1α w α wrand w  r ≡ w  rmod k with 1 ≤ w  r≤ k.

Proof Let

Lt  α1A1t  A2t  · · ·  A k t  α2A2t  · · ·  A k t  A1t

 · · ·  α k A k t  A1t  · · ·  A k−1t. 3.7

By taking indices “modk” and using2.20 ofCorollary 2.6follows that

Lt ∗ L T t  α2

1

A1t ∗ A T

1t · · · A k t ∗ A T

 · · ·  α2

k

A k t ∗ A T

k tA1t ∗ A T

1t

 · · · A k−1t ∗ A T

k

i /  j

α i α j

A i t ∗ A T

j tA j1t ∗ A T

j1t

 · · · A j−1t ∗ A T

j−1t.

3.8

Now on usingCorollary 2.6and the commutativity of Hadamard convolution product yields

P km T t ∗ Lt ∗ L T t ∗ P km t 

k



i1

α2i

k



i1

•A i t ∗ A T



k−1



r1

μ r k



•A w t ∗ A T

wrt

3.9

where μ r  k

w α w α wr and w  r ≡ w  r mod k with 1 ≤ w  r≤ k then

P T

km t ∗ Lt ∗ P kn t α1P T

km t ∗ A1t  A2t  · · ·  A k t ∗ P kn t

 α2P T

km t ∗ A2t  · · ·  A k t  A1t ∗ P kn t

 · · ·  α k P km T t ∗ A k t  A1t  · · ·  A k−1t ∗ P kn t



k



i1

α i k



i1

•A i t

3.10

Thus it follows that



P T

km t ∗ Lt ∗ P kn tT 

k

i1•A i t

T

,

P km T t∗Lt∗P kn t∗P km T t∗Lt∗P kn tT 

k

i1

α i

2 k

i1

•A i t ∗

k

i1

•A i t

T

.

3.11

Now by applying inequality2.26, and 3.6 and 3.7 thus we establishTheorem 3.2

We note that many special cases can be derived fromTheorem 3.2 For example, in order to see that inequality3.6 is an extension of inequality 3.2 we set α1  1 and α2 · · ·  α k 0 Next, we recover inequality3.1 ofTheorem 3.1, by letting k  2, then μ1  2

w1α w α w1

with w  1 ≡ w  1mod 2, that is, μ1 2α1α2then we have

α21 α2

2

A1t ∗ A T

1t•A2t ∗ A T

2t  2α1α2

A1t ∗ A T

2t•A2t ∗ A T

1t

≥ α1 α22A1t•At ∗ A1t•A2t T

3.12

By simplification we have

A

1t ∗ A T

1t•A2t ∗ A T

2t s A1t ∗ A T

2t•A2t ∗ A T

1t

≥ 1  sA1t•A2t ∗ A1t•A2t T

3.13

for every s ∈ −1, 1, just as required Finally, if we let k  3, α1 1, and α2  α3 −1/2, then

on usingTheorem 3.2we have an attractive inequality as follows

A1t ∗ A T

1t•A2t ∗ A T

2t•A3t ∗ A T

3t

≥ 1 2



A1

t ∗ A T

2t•A2t ∗ A T

3t•A3t ∗ A T

1t

A2t ∗ A T

1t•A3t ∗ A T

2t•A1t ∗ A T

3t.

3.14

Acknowledgments

The authors gratefully acknowledge that this research partially supported by Ministry of Science, Technology and InnovationsMOSTI, 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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We note that Hadamard convolution product differs from the convolution product of matrices in many ways One important difference is the commutativity of Hadamard. .. class="text_page_counter">Trang 8

Further,Theorem 3.1can be extended to the case of Hadamard convolution products which involves finite number of matrices. ..

Proof The proof is by induction on k If k  2, then the result is true by using 2.17 Now

suppose that corollary holds for the Hadamard convolution product of k matrices Then