Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 736243, 10 pages doi:10.1155/2009/736243 ResearchArticleOntheConnectionbetweenKroneckerandHadamardConvolutionProductsofMatricesandSome Applications Adem Kılıc¸man 1 and Zeyad Al Zhour 2 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 KroneckerandHadamardconvolutionproductsand present some important connections between these two products. Further we establish some attractive inequalities for Hadamardconvolution product. It is also proved that the results can be extended to the finite number of matrices, andsome basic properties of matrix convolutionproducts are also derived. Copyright q 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 theConvolution 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, theconvolution 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 ofmatricesand presented a new method to obtain closed form solutions of transition probabilities and dependability measures and then solved the renewal matrix equation by using theconvolution 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 ofthe autocovariances matrix function can be expressed in terms oftheKroneckerconvolution product. Recently in 1, Kilic¸man and Al Zhour 2 Journal of Inequalities and Applications presented the iterative solution of such coupled matrix equations based ontheKroneckerconvolution structures. In this paper, we consider KroneckerandHadamardconvolutionproducts for matricesand define the so-called Dirac identity matrix D n t which behaves like a group identity element under theconvolution matrix operation. Further, we present some results which includes matrix equalities as well as inequalities related to these productsand give attractive application to the inequalities that involves Hadamardconvolution 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 tδtI n are the Dirac delta function and Dirac identity matrix, respectively; here, the notation I n is the scalar identity matrix of order n×n. The notations At ∗ Bt, At Bt,andAt•Bt are convolution product, Kroneckerconvolution product andHadamardconvolution product of matrix functions At and Bt, respectively. 2. Matrix ConvolutionProductsandSome Properties In this section, we introduce KroneckerandHadamardconvolutionproductsof matrices, obtain some new results, and establish connections between these products that will be useful in some applications. Definition 2.1. Let Atf ij t ∈ M I m,n , Btg jr t ∈ M I n,p ,andCtz ij t ∈ M I m,n . The convolution, KroneckerconvolutionandHadamardconvolutionproducts are matrix functions defined for t ≥ 0 as follows whenever the integral is defined. i Convolution product A t ∗ B t h ir t with h ir t n k1 t 0 f ik t − x g kr x dx n k1 f ik t ∗ g kr t . 2.1 ii Kroneckerconvolution product A t B t f ij t ∗ B t ij . 2.2 iii Hadamardconvolution product A t •C t f ij t ∗ z ij t ij . 2.3 where f ij t ∗ Bt is the ijth submatrix of order n × p;thusAt 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 oftheconvolution product andKronecker product of matrices, respectively. Journal of Inequalities and Applications 3 Theorem 2.2. Let At, Bt, Ct ∈ M I n , and let D n tδtI n ∈ M I n . Then for scalars α and β i αA t βB t ∗ C t α A t ∗ C t β B t ∗ C t , 2.4 ii A t ∗ B t ∗ C t A t ∗ B t ∗ C t , 2.5 iii A t ∗ D n t D n t ∗ A t A t , 2.6 iv A t ∗ B t 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 tδtI n ∈ M I n .Then i D n t A t diag A t ,A t , ,A t , 2.8 ii D n t D m t D nm t , 2.9 iii A t C t B t A t B t C t B t , 2.10 iv At Bt T A T t B T t , 2.11 v A t B t ∗ C t D t A t ∗ C t B t ∗ D t , 2.12 vi A t D m t ∗ D n t B t D n t B t ∗ A t D m t A t B t . 2.13 4 Journal of Inequalities and Applications The above results can easily be extended to the finite number ofmatrices 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 of Theorem 2.3v. Now we can proceed by induction on k. Assume that Corollary 2.4 holds for productsof k − 1 matrices. Then A 1 t B 1 t ∗ A 2 t B 2 t ∗···∗ A k t B k t { A 1 t B 1 t ∗ A 2 t B 2 t ∗···∗ A k−1 t B k−1 t } ∗ A k t B k t { A 1 t ∗ A 2 t ∗···∗A k−1 t B 1 t ∗ B 2 t ∗···∗B k−1 t } ∗ A k t B k t { A 1 t ∗ A 2 t ∗···∗A k−1 t ∗ A k t } { B 1 t ∗ B 2 t ∗···∗B k−1 t ∗ B k t } k i1 ∗ A i t k i1 ∗ B i t . 2.16 Similarly we can prove ii. Theorem 2.5. Let Atf ij t, and let Btg ij t ∈ M I m,n .Then A•B t P T m t ∗ A B t ∗ P n t . 2.17 Here, P n tVe c E n 11 t, ,Vec E n nn t ∈ M n 2 ,n and E ij te i t ∗ e T j t of order n × n, e i t is the ith column of Dirac identity matrix D n tδtI n ∈ M n with property P T n t ∗P n tD n t. In particular, if m n, then we have A•B t P T n t ∗ A B t ∗ P n t . 2.18 Journal of Inequalities and Applications 5 Proof. Compute P T m t ∗ A B t ∗ P n t Vec E m 11 t, ,Vec E m mm t T ∗ A B t ∗ Vec E n 11 t , ,Vec E n nn t n k1 diag f ik t ,f 2k t , ,f mk t ∗ B t ∗ E n kk t n k1 f ik t ∗ g ij t ∗ δ jk t f ij t ∗ g ij t A•B t . 2.19 This completes the proof of Theorem 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 T km t E m 11 t , 0 m , ,0 m ,E m 22 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 e ntries 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 s1 m s zero matrices 0 m between E m ii t and E m i1,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.Ifk 2, then the result is true by using 2.17.Now suppose that corollary holds for theHadamardconvolution product of k matrices. Then we have k1 i1 •A i t A 1 t • k1 i1 •A i t P T m t ∗ A 1 t k1 i1 •A i t ∗ P n t P T m t ∗ D m t P T km t ∗ k1 i1 A i t ∗ D n t P kn t ∗ P n t P T m t ∗ D m t P T km t ∗ k1 i1 A i t ∗ D n t P kn t ∗ P n t , 2.23 6 Journal of Inequalities and Applications which is based onthe fact that P T m t ∗ D m t P T km t P T k1 m 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.ThenP T m t ∗ P m tD m t and P m ∗ P T m is a diagonal m 2 × m 2 matrix of zeros, and then the following inequality satisfied 0 ≤ P m t ∗ P T m t ≤ D m 2 . 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 m 2 × n 2 matrix Lt, P T m t ∗ L t ∗ L T t ∗ P m t ≥ P T m t ∗ L t ∗ P n t ∗ P T m t ∗ Lt ∗ P n t T ≥ 0. 2.26 Proof. By Corollary 2.7, it is clear that D n 2 t ≥ P n t ∗ P T n t ≥ 0andso P T m t ∗ L t ∗ D n 2 t ∗ L T t ∗ P m t P T m t ∗ L t ∗ L T t ∗ P m t ≥ P T m t ∗ L t ∗ P n t ∗ P T n t ∗ L T t ∗ P m t P T m t ∗ L t ∗ P n t ∗ P T m t ∗ Lt ∗ P n t T ≥ 0. 2.27 This completes the proof of Theorem 2.8. We note that Hadamardconvolution product differs from theconvolution product ofmatrices in many ways. One important difference is the commutativity ofHadamardconvolution multiplication A•B t B•A t . 2.28 Similarly, the diagonal matrix function can be formed by using Hadamardconvolution multiplication with Dirac identity matrix. For example, if At, Bt ∈ M I n , and D n t Dirac identity then we have i A•BtA ∗ Bt if and only if At and Bt are both diagonal matrices; iiA•Bt•D n tA•D n t ∗ B•D n t. Journal of Inequalities and Applications 7 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 ofHadamardconvolution 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 s A t •B t ∗ At•Bt T . 3.1 In particular, if s 0, then we have A t ∗ A T t • B t ∗ B T t ≥ A t •B t ∗ At•Bt T . 3.2 Proof. Choose LtαAt BtβBt At, where At,andBt ∈ M I m,n and α, β are real scalars not both zero. Since L t ∗ L T t αA t B t βB t A t ∗ αA t B t βB t A t T , 3.3 on using Theorem 2.5 we can easily obtain that P T m t ∗ L t ∗ L T t ∗ P m t α 2 A t ∗ A T t • B t ∗ B T t αβ A t ∗ B T t • B t ∗ A T t αβ B t ∗ A T t • A t ∗ B T t β 2 B t ∗ B T t • A t ∗ A T t α 2 β 2 A t ∗ A T t • B t ∗ B T t 2αβ A t ∗ B T t • B t ∗ A T t . 3.4 Now one can also easily show that P T m t ∗ L t ∗ P n t ∗ P T m t ∗ Lt ∗ P n t T α β 2 A t •B t ∗ At•Bt T . 3.5 By setting s 2αβ/α 2 β 2 , then it follows that s1 α β 2 /α 2 β 2 ; further the arithmetic- geometric mean inequality ensures that |s|≤1 andthe choices β 1andα ∈ −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. 8 Journal of Inequalities and Applications Further, Theorem 3.1 can be extended to the case ofHadamardconvolutionproducts which involves finite number ofmatrices as follows. Theorem 3.2. Let A i ∈ M I m,n 1 ≤ i ≤ k, k ≥ 2. Then for real scalars α 1 ,α 2 , , α k , which are not all zero k i1 α 2 i k i1 • A i t ∗ A T i t k−1 r1 μ r k w1 • A w t ∗ A T wr t ≥ k i1 α i 2 k i1 •A i t k i1 •A i t T , 3.6 where μ r k w1 α w α wr and w r ≡ w r mod k with 1 ≤ w r ≤ k. Proof. Let L t α 1 A 1 t A 2 t ···A k t α 2 A 2 t ···A k t A 1 t ··· α k A k t A 1 t ···A k−1 t . 3.7 By taking indices “modk”andusing2.20 of Corollary 2.6 follows that L t ∗ L T t α 2 1 A 1 t ∗ A T 1 t ··· A k t ∗ A T k t ··· α 2 k A k t ∗ A T k t A 1 t ∗ A T 1 t ··· A k−1 t ∗ A T k−1 t k i / j α i α j A i t ∗ A T j t A j1 t ∗ A T j1 t ··· A j−1 t ∗ A T j−1 t . 3.8 Now on using Corollary 2.6 andthe commutativity ofHadamardconvolution product yields P T km t ∗ L t ∗ L T t ∗ P km t k i1 α 2 i k i1 • A i t ∗ A T i t k−1 r1 μ r k w1 • A w t ∗ A T wr t 3.9 Journal of Inequalities and Applications 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 ∗ L t ∗ P kn t α 1 P T km t ∗ A 1 t A 2 t ···A k t ∗ P kn t α 2 P T km t ∗ A 2 t ···A k t A 1 t ∗ P kn t ··· α k P T km t ∗ A k t A 1 t ···A k−1 t ∗ 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 t T k i1 α i k i1 •A i t T , P T km t ∗L t ∗P kn t ∗ P T km 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 inequality 2.26,and3.6 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 α 1 1andα 2 ··· α k 0. Next, we recover inequality 3.1 of Theorem 3.1, by letting k 2, then μ 1 2 w1 α w α w1 with w 1 ≡ w 1 mod 2, that is, μ 1 2α 1 α 2 then we have α 2 1 α 2 2 A 1 t ∗ A T 1 t • A 2 t ∗ A T 2 t 2α 1 α 2 A 1 t ∗ A T 2 t • A 2 t ∗ A T 1 t ≥ α 1 α 2 2 A 1 t •A t ∗ A 1 t•A 2 t T . 3.12 By simplification we have A 1 t ∗ A T 1 t • A 2 t ∗ A T 2 t s A 1 t ∗ A T 2 t • A 2 t ∗ A T 1 t ≥ 1 s A 1 t •A 2 t ∗ A 1 t•A 2 t T 3.13 10 Journal of Inequalities and Applications for every s ∈ −1, 1, just as required. Finally, if we let k 3, α 1 1, and α 2 α 3 −1/2, then on using Theorem 3.2 we have an attractive inequality as follows. A 1 t ∗ A T 1 t •A 2 t ∗ A T 2 t •A 3 t ∗ A T 3 t ≥ 1 2 A 1 t ∗ A T 2 t • A 2 t ∗ A T 3 t • A 3 t ∗ A T 1 t A 2 t ∗ A T 1 t • A 3 t ∗ A T 2 t • A 1 t ∗ A T 3 t . 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. References 1 A. Kilic¸man and Z. Al Zhour, “Iterative solutions of coupled matrix convolution equations,” Soochow Journal of Mathematics, vol. 33, no. 1, pp. 167–180, 2007. 2 N. Limnios, “Dependability analysis of semi-Markov systems,” Reliability Engineering and System Safety, vol. 55, no. 3, pp. 203–207, 1997. 3 S. Saitoh, “New norm type inequalities for linear mappings,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 3, article 57, pp. 1–5, 2003. 4 S. Saitoh, V. 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Boshnakov, “The asymptotic covariance matrix ofthe multivariate serial correlations,” Stochastic Processes and Their Applications, vol. 65, no. 2, pp. 251–258, 1996. . 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. Corporation Journal of Inequalities and Applications Volume 2009, Article ID 736243, 10 pages doi:10.1155/2009/736243 Research Article On the Connection between Kronecker and Hadamard Convolution Products. 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