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

Volume 2009, Article ID 736243,10pages doi:10.1155/2009/736243

Research Article

On the Connection between Kronecker and

Hadamard Convolution Products of Matrices

and Some Applications

Adem Kılıc¸man

_{and Zeyad Al Zhour}

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,[email protected]

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.

Copyrightq2009 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

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 Dnt 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 MI

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

t ≥ 0, and if m n, we writeMI

n instead of Mm,nI . The notationATtis the transpose of matrix functionAt. The notationsδtandDnt δtInare the Dirac delta function and Dirac identity matrix, respectively; here, the notationInis the scalar identity matrix of order

n×n. The notationsAt∗Bt,AtBt, andAt•Btare convolution product, Kronecker convolution product and Hadamard convolution product of matrix functionsAtandBt, 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. LetAt fijt∈MIm,n,Bt gjrt∈MIn,p, andCt zijt∈Mm,nI . The convolution, Kronecker convolution and Hadamard convolution products are matrix functions defined fort≥0 as followswhenever the integral is defined.

iConvolution product

At∗Bt hirt withhirt n

k1 _t

0

fikt−xgkrxdx n

k1

fikt∗gkrt. 2.1

iiKronecker convolution product

AtBt fijt∗Bt

ij. 2.2

iiiHadamard convolution product

At•Ct fijt∗zijt

ij. 2.3

wherefijt∗Btis theijth submatrix of ordern×p; thusAtBtis of ordermn×np,

At∗Btis of orderm×p, and similarly, the productAt•Ctis of orderm×n.

Theorem 2.2. LetAt,Bt, Ct∈MI

n, and letDnt δtIn∈MIn. Then for scalarsαandβ

i

αAt βBt∗Ct αAt∗Ct βBt∗Ct, 2.4

ii

At∗Bt∗Ct At∗Bt∗Ct, 2.5

iii

At∗Dnt Dnt∗At At, 2.6

iv

At∗BtT BTt∗ATt. 2.7

Theorem 2.3. LetAt, Ct∈MI

m,n,Bt∈MIp,q,and letDnt δtIn∈MIn. Then

i

DntAt diagAt, At, . . . , At, 2.8

ii

DntDmt Dnmt, 2.9

iii

At CtBt AtBt CtBt, 2.10

iv

AtBtT ATtBTt, 2.11

v

AtBt∗CtDt At∗CtBt∗Dt, 2.12

vi

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

Corollary 2.4. LetAitandBit∈MIn1≤i≤kbe matrices. Then

i

k

i1

∗AitBit

k

i1

∗Ait

k

i1 ∗Bit

, 2.14

ii

k

i1

Ait∗Bit

k

i1

Ait

∗ k

i1 Bit

. 2.15

Proof. i The proof is a consequence ofTheorem 2.3v. Now we can proceed by induction onk. Assume thatCorollary 2.4holds for products ofk−1 matrices. Then

A1tB1t∗A2tB2t∗ · · · ∗AktBkt

{A1tB1t∗A2tB2t∗ · · · ∗Ak−1tBk−1t} ∗AktBkt

{A1t∗A2t∗ · · · ∗Ak−1tB1t∗B2t∗ · · · ∗Bk−1t} ∗AktBkt

{A1t∗A2t∗ · · · ∗Ak−1t∗Akt} {B1t∗B2t∗ · · · ∗Bk−1t∗Bkt}

k

i1 ∗Ait

k

i1 ∗Bit

.

2.16

Similarly we can proveii.

Theorem 2.5. LetAt fijt, and letBt gijt∈Mm,nI . Then

A•Bt P_mTt∗ABt∗Pnt. 2.17

Here,Pnt VecE₁₁nt, . . . ,VecEnnnt∈Mn2_,nandE_ijt e_it∗eT

jtof ordern×n,eit

is theith column of Dirac identity matrixDnt δtIn∈Mnwith propertyPnTt∗Pnt Dnt.

In particular, ifmn, then we have

Proof. Compute

P_mTt∗ABt∗Pnt

VecE₁₁mt, . . . ,VecEmmmt

T

∗ABt

∗VecE₁₁nt, . . . ,VecEnnnt

n

k1

diagfikt, f2kt, . . . , fmkt

∗Bt∗E_kknt

n

k1

fikt∗gijt∗δjkt

fijt∗gijt

A•Bt.

2.19

This completes the proof ofTheorem 2.5.

Corollary 2.6. LetAit∈MIm,n1≤i≤k, k ≥2. Then there exist two matricesPkmtof order

mk_×mandP

kntof ordernk×nsuch that k

i1

•Ait PkmT t∗ k

i1

Ait

∗Pknt, 2.20

where

P_kmT t E₁₁mt,0m, . . . ,0m, E₂₂mt,0m, . . . ,0m, Emmmt

2.21

is of orderm×mk_,0m_{is anm×mmatrix with all entries equal to zero,E}m

ij tis anm×mmatrix of

zeros except for aδtin theijth position, and there arek_s−₁2ms_{zero matrices0}m_betweenEm

ii t

andE_im_1,i1t(1≤i≤m−1). In particular, ifmn, then we have

k

i1

•Ait PkmT t∗ k

i1

Ait

∗Pkmt. 2.22

Proof. The proof is by induction onk. Ifk 2, then the result is true by using2.17. Now suppose that corollary holds for the Hadamard convolution product ofkmatrices. Then we have

k1

i1

•Ait A1t•

k1

i1 •Ait

P_mTt∗ A1t

k1

i1 •Ait

∗Pnt

P_mTt∗ DmtP_kmT t

∗ k1

i1

Ait

∗DntPknt

∗Pnt

P_mTt∗DmtP_kmT t

∗ k1

i1

Ait

∗DntPknt∗Pnt,

which is based on the fact that

PT mt∗

DmtP_kmT t

PT

k1mt, DntPknt∗Pnt Pk1nt, 2.24

and thus the inductive step is completed.

Corollary 2.7. LetAt, Bt ∈ MI

m and Pmtbe a matrix of zeros andDmtthat satisfies the

2.17. ThenPT

mt∗Pmt DmtandPm∗PmT is a diagonalm2×m2matrix of zeros, and then the

following inequality satisfied

0≤Pmt∗PmTt≤Dm2. 2.25

Proof. It follows immediately by the definition of matrixPmt.

Theorem 2.8. LetAtandBt∈MI

m,n. Then for anym2×n2matrixLt,

P_mTt∗Lt∗LTt∗Pmt≥

P_mTt∗Lt∗Pnt

∗P_mTt∗Lt∗Pnt

T

≥0. 2.26

Proof. ByCorollary 2.7, it is clear thatDn2t≥P_nt∗P_nTt≥0 and so

P_mTt∗Lt∗Dn2t∗LTt∗P_mt P_mTt∗Lt∗LTt∗P_mt

≥P_mTt∗Lt∗Pnt∗PnTt∗LTt∗Pmt

P_mTt∗Lt∗Pnt

∗P_mTt∗Lt∗Pnt

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, ifAt,Bt ∈MI

n,andDntDirac identity then we have

iA•Bt A∗Btif and only ifAtandBtare both diagonal matrices;

3. Some New Applications

Now based on inequality2.26in 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 ∈ MI

m,n and for s ∈ −1,1, we have At ∗

AT_t•Bt∗BT_{t sAt∗B}T_t•Bt∗AT_t

≥1sAt•Bt∗At•BtT. 3.1

In particular, ifs0, then we have

At∗ATt•Bt∗BTt≥At•Bt∗At•BtT. 3.2

Proof. ChooseLt αAtBt βBtAt, whereAt, andBt∈MI

m,nandα, βare real scalars not both zero. Since

Lt∗LTt αAtBt βBtAt∗αAtBt βBtAtT, 3.3

on usingTheorem 2.5we can easily obtain that

P_mTt∗Lt∗LTt∗Pmt

α2At∗ATt•Bt∗BTt αβAt∗BTt•Bt∗ATt αβBt∗ATt•At∗BTt β2Bt∗BTt•At∗ATt α2_β2_At∗AT_t•Bt∗BT_t

2αβAt∗BTt•Bt∗ATt.

3.4

Now one can also easily show that

P_mTt∗Lt∗Pnt

∗P_mTt∗Lt∗Pnt

T

αβ2At•Bt∗At•BtT. 3.5

By settings2αβ/α2_β2_{, then it follows thats1 αβ}2_/α2_β2_{; further the}

arithmetic-geometric mean inequality ensures that|s| ≤ 1 and the choicesβ 1 andα∈−1,1thuss

takes all values in−1,1. Now by using 3.4,3.5and inequality2.26we can establish

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

Theorem 3.2. LetAi∈MIm,n1≤i≤k, k≥2. Then for real scalarsα1, α2, . . . , αk, which are not

all zero

k

i1

α2_i

k

i1

•Ait∗ATit

k−1

r1

μr k

w1

•Awt∗AT_wrt

≥ k

i1

αi

2 _k

i1 •Ait

k

i1 •Ait

T

,

3.6

whereμr

_k

w1αwαwrandwr≡wrmodkwith1≤wr≤k.

Proof. Let

Lt α1A1tA2t · · · Akt α2A2t · · · AktA1t

· · ·αkAktA1t · · · Ak−1t.

3.7

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

Lt∗LTt α2₁A1t∗AT1t

· · · Akt∗AT_kt

· · ·α2_kAkt∗AT_kt

A1t∗AT1t

· · · Ak−1t∗AT_k−1t

k

i /j

αiαj

Ait∗ATjt

Aj1t∗ATj1t

· · · Aj−1t∗ATj−1t

.

3.8

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

P_kmT t∗Lt∗LTt∗Pkmt k

i1

α2_i

k

i1

•Ait∗AT_it

k−1

r1

μr k

w1

•Awt∗AT_wrt

whereμr kwαwα_wr andwr ≡wr

modkwith 1≤wr≤kthen

PT

kmt∗Lt∗Pknt

α1P_kmT t∗A1tA2t · · · Akt∗Pknt

α2P_kmT t∗A2t · · · AktA1t∗Pknt

· · ·αkPkmT t∗AktA1t · · · Ak−1t∗Pknt

k

i1

αi k

i1 •Ait

.

3.10

Thus it follows that

PT

kmt∗Lt∗Pknt

T k

i1

αi k

i1 •Ait

T

,

P_kmT t∗Lt∗Pknt

∗P_kmT t∗Lt∗Pknt

T

k

i1

αi

2 _k

i1 •Ait

∗ k

i1 •Ait

T

.

3.11

Now by applying inequality2.26, and3.6and3.7thus we establishTheorem 3.2. We note that many special cases can be derived fromTheorem 3.2. For example, in order to see that inequality3.6is an extension of inequality3.2we setα1 1 andα2· · ·αk0. Next, we recover inequality3.1ofTheorem 3.1, by lettingk 2, thenμ1

₂

w1αwαw1

withw1≡w1mod 2, that is,μ12α1α2then we have

α2₁α2₂A1t∗AT₁t

•A2t∗AT2t

2α1α2

A1t∗AT2t

•A2t∗AT₁t

≥α1α22A1t•At∗A1t•A2tT.

3.12

By simplification we have

A1t∗AT1t

•A2t∗AT2t

sA1t∗AT2t

•A2t∗AT1t

≥1sA1t•A2t∗A1t•A2tT

for everys∈−1,1, just as required. Finally, if we letk3,α11, andα2 α3−1/2, then

on usingTheorem 3.2we have an attractive inequality as follows.

A1t∗AT1t

•A2t∗AT2t

•A3t∗AT3t

≥ 1

2

A1

t∗AT₂t•A2t∗AT3t

•A3t∗AT1t

A2t∗AT1t

•A3t∗AT2t

•A1t∗AT3t

.

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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