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Some new connections between matrix products for partitioned and non-partitioned matrices. (English) Zbl 1146.15014

The authors gather several known matrix products (Kronecker, Hadamard, Tracy-Singh, Khatri-Rao) and collect into one paper several known connections between these products.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
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[1] Al Zhour, Z.; Kiliçman, A., Matrix equalities and inequalities involving Khatri-Rao and Tracy-Singh sums, J. Inequal. Pure Appl. Math., 7, 1, 496-513 (2006), Article 34 · Zbl 1132.15018
[2] Al Zhour, Z.; Kiliçman, A., Extensions and generalization inequalities involving the Khatri-Rao product of several positive matrices, J. Inequal. Appl., 2006, 1-21 (2006), Article ID 80878 · Zbl 1134.15012
[3] Graham, A., Kronecker Products and Matrix Calculus with Applications (1981), Ellis Horwood Ltd.: Ellis Horwood Ltd. New York · Zbl 0497.26005
[4] Lev-Ari, H., Efficient solution of linear matrix equations with application to multistatic antenna array processing, Commun. Inf. Syst, 5, 1, 123-130 (2005) · Zbl 1116.65050
[5] Liu, S., Several inequalities involving Khatri-Rao products of positive semi definite matrices, Linear Algebra Appl., 345, 175-186 (2002) · Zbl 1016.15016
[6] Rao, C. R.; Rao, M. B., Matrix Algebra and its Applications to Statistics and Econometrics (1998), World Scientific Pub. Co. Pte. Ltd.: World Scientific Pub. Co. Pte. Ltd. Singapore · Zbl 0915.15001
[7] Steeb, W.-H., Matrix Calculus and Kronecker Product with Applications and C++ Programs (1997), World Scientific Pub. Co. Pte. Ltd.: World Scientific Pub. Co. Pte. Ltd. Singapore
[8] Visick, G., A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product, Linear Algebra Appl., 304, 45-68 (2000) · Zbl 0946.15015
[9] Zhang, F., Matrix Theory: Basic Results and Techniques (1999), Springer-Verlag: Springer-Verlag New York Inc. · Zbl 0948.15001
[10] MacRae, E. C., Matrix derivatives with an application to an adaptive linear decision problem, Ann. Statist., 2, 337-364 (1974) · Zbl 0285.26013
[11] Al Zhour, Z.; Kiliçman, A., New algebraic method for solving the axial \(N\)-index transportation problem based on the Kronecker product, Matematika, 21, 2, 113-123 (2005))
[12] Al Zhour, Z.; Kiliçman, A.; Abu Hasan, M., A new algebraic method for solving the balanced transportation problem based on the Kronecker product, IRSIAM J. Appl. Math., 1, 1, 64-80 (2005)
[13] Ding, F.; Chen, T., Iterative least-squares solutions of coupled Sylvester matrix equations, Syst. Control Lett., 54, 95-107 (2005) · Zbl 1129.65306
[14] Kiliçman, A.; Al Zhour, Z., The general common exact solutions of coupled linear matrix and matrix differential equations, J. Anal. Comput., 1, 1, 15-29 (2005) · Zbl 1121.34303
[15] Kiliçman, A.; Al Zhour, Z., Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput., 187, 1, 250-265 (2007) · Zbl 1123.65063
[16] A. Kiliçman, Z. Al Zhour, Iterative solutions of coupled matrix convolution equations, Soochow J. Math. (2007) (in press). Corrected proof, available online at http://journal.math.scu.edu.tw/mp/scu/; A. Kiliçman, Z. Al Zhour, Iterative solutions of coupled matrix convolution equations, Soochow J. Math. (2007) (in press). Corrected proof, available online at http://journal.math.scu.edu.tw/mp/scu/
[17] Kiliçman, A.; Al Zhour, Z., Improvements on geometric means related to the Tracy-Singh products of positive matrices, Matematika, 21, 2, 49-65 (2005)
[18] A. Kiliçman, Z. Al Zhour, Connection between the Kronecker and Hadamard convolution products of matrices and some applications, Int. J. Pure Appl. Math. (2007) (in press); A. Kiliçman, Z. Al Zhour, Connection between the Kronecker and Hadamard convolution products of matrices and some applications, Int. J. Pure Appl. Math. (2007) (in press)
[19] Tauber, S., An applications of the Hadamard product to air pollution, Appl. Math. Comput., 4, 167-176 (1978) · Zbl 0383.35032
[20] A. Kiliçman, Z. Al Zhour, Vector least-squares solutions of coupled singular matrix equations, J. Comput. Appl. Math. (2007) (in press). Corrected proof, available online 29 November 2006 at http://www.sciencedirect.com; A. Kiliçman, Z. Al Zhour, Vector least-squares solutions of coupled singular matrix equations, J. Comput. Appl. Math. (2007) (in press). Corrected proof, available online 29 November 2006 at http://www.sciencedirect.com
[21] Van Loan, G. F., The ubiquitous Kronecker product, J. Comput. Appl. Math., 123, 85-100 (2000) · Zbl 0966.65039
[22] Z. Al Zhour, A. Kiliçman, Some topics on weighted generalized inverses and Kronecker product of matrices, Malaysian J. Math. Sci. (2007) (in press); Z. Al Zhour, A. Kiliçman, Some topics on weighted generalized inverses and Kronecker product of matrices, Malaysian J. Math. Sci. (2007) (in press)
[23] Horn, R. A.; Johnson, C., Topics in Matrix Analysis (1990), Cambridge University Press: Cambridge University Press New York
[24] Xu, L.; Stoica, P.; Li, J., A block-diagonal growth curve model, Digital Signal Processing, 16, 6, 902-912 (2006)
[25] Brewer, J. W., Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits Syst., CAS-25, 9, 772-781 (1978) · Zbl 0397.93009
[26] Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26, 203-241 (1979) · Zbl 0495.15018
[27] Tracy, D. S.; Jinadasa, K. G., Partitioned Kronecker products of matrices and applications, Canad. J. Statist., 17, 107-120 (1989) · Zbl 0684.62044
[28] Mićić, J.; Seo, Y.; Takahasi, S.-E.; Tominaga, M., Inequalities of Furuta and Mond-Pečarić, Math. Inequal. Appl., 2, 1, 83-111 (1999) · Zbl 0924.47013
[29] Khatri, C. G.; Rao, C. R., Solution to some functional equations and their applications to characterization of probability distributions, Sankhya, 30, 167-180 (1968) · Zbl 0176.49004
[30] Tracy, D. S.; Singh, R. P., A new matrix product and its applications in partitioned matrix differentiation, Statist. Neerlandica, 26, 143-157 (1972) · Zbl 0267.15009
[31] Johnson, C. R.; Nylen, P., Largest singular value submultiplicativity, SIAM J. Matrix Anal. Appl., 12, 1-6 (1991) · Zbl 0721.15011
[32] Koning, R. H.; Neudecker, H.; Wasbeek, T., Block Kronecker products and the vec-operator, Linear Algebra Appl., 149, 267-277 (1991)
[33] Magnus, J. R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Statist, 7, 381-394 (1979) · Zbl 0414.62040
[34] Liu, S., Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra. Appl., 289, 267-277 (1999) · Zbl 0937.15015
[35] Wei, Y.; Zhang, F., Equivalence of a matrix product to the Kronecekr product, Hadronic J. Suppl., 15, 3, 327-331 (2000) · Zbl 1066.15504
[36] Cao, C.-G.; Zhang, X.; Yang, Z.-P., Some inequalities for the Khatri-Rao product of matrices, Electron. J. Linear Algebra, 9, 276-281 (2002) · Zbl 1020.15021
[37] Tracy, D. S., Balanced partitioned matrices and their Kronecker products, Comput. Statist. Data Anal., 10, 315-323 (1990) · Zbl 0825.62549
[38] Zhang, X.; Yang, Z.-P.; Cao, C.-G., Inequalities involving Khatri-Rao products of positive semi-definite matrices, Appl. Math. E-Notes, 2, 117-124 (2002) · Zbl 0998.15026
[39] Horn, R. A.; Mathias, R., Block-matrix generalizations of Shur’s basic theorems on Hadamard products, Linear Algebra Appl., 172, 337-346 (1992) · Zbl 0770.15007
[40] Zhang, F., Schur complements and matrix inequalities in the lowner ordering, Linear Algebra Appl., 321, 399-410 (2000) · Zbl 0977.15008
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