Operator inequalities involving Khatri-Rao Sums and Moore-Penrose inverses
DOI:
https://doi.org/10.11113/mjfas.v0n0.711Keywords:
tensor product, Khatri-Rao product (sum), Tracy-Singh product (sum), Moore-Penrose inverse, operator inequalityAbstract
We establish relations between the Khatri-Rao sum of Hilbert space operators and ordinary products, powers, ordinary inverses, and Moore-Penrose inverses in terms of inequalities. In particular, such relations hold for the tensor sum of operators and the Khatri-Rao sum of complex matrices.
References
Tracy, D.S. and Singh, R.P. 1972. A new matrix product and its applications in partitioned matrix differentiation. Statistica Neerlandica. 26(4), 143-157.
Khatri, C.G. and Rao, C.R. 1968. Solutions to some functional equations and their applications to characterization of probability distributions. Sankhya. 26(4), 167-180.
Al Zhour, Z.A. and Kilicman, A. 2006. Extension and generalization inequalities involving the Khatri-Rao product of several positive matrices, Journal of Inequalities and Applications, 1-21.
Liu, S. and Trenkler, G. 2008. Hadamard, Khatri-Rao, Kronecker and other matrix products, International Journal of Information and Systems Sciences, 4(1), 160-177.
Shuangzhe, L. 1999. Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra Appl. 289(1-3), 267-277.
Van Loan, C. F. 2000. The ubiquitous Kronecker product, International Journal of Computational and Applied Mathematics.123, 85-100.
Al Zhour, Z.A. and Kilicman, A. 2006. Matrix equalities and inequalities involving Khatri-Rao and Tracy-Singh sums, Journal of Inequalities in Pure and Applied Mathematics. 7(1), 11-17.
Kubrusly C.S. and Levan N. 2011. Preservation of tensor sum and tensor product, Acta Mathematica Universitatis Comenianae. 80(1), 133-142.
Ploymukda, A. Chansangiam, P. and W. Lewkeeratiyutkul. 2017. Algebraic and order properties of Tracy-Singh products for operator matrices, Journal of Computational Analysis and Applications, 24(4), 656-664.
Ploymukda, A. Chansangiam, P. and W. Lewkeeratiyutkul. 2017. Analytic properties of Tracy-Singh products for operator matrices, Journal of Computational Analysis and Applications, 24(4), 665-674
Ploymukda, A. and Chansangiam, 2016. P. Khatri-Rao products for operator matrices acting on the direct sum of Hilbert spaces, Journal of Mathematics, Article ID 8301709, 7 pages. http://dx.doi.org/10.1155/2016/8301709
Ploymukda, A. and Chansangiam, P. 2017. Several inequalities for Khatri-Rao products of Hilbert space operators, Communications in Mathematics and Applications. 8(1), 1-15.
Ploymukda, A. and Chansangiam, P. 2017. Khatri-Rao sums for Hilbert space operators, Songklanakarin Journal of Science and Technology, accepted.
Penrose, R. 1955. A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51, 406-413.
Caradus, S.R. Generalized Inverses and Operator Theory, Queen's Papers in Pure and Applied Mathematics no. 50, Queen's University, Kingston, Ontario, 1978.
Xu, Q. and Sheng, L. 2008. Positive semi-definite matrices of adjointable operators on Hilbert-modules, Linear Algebra and its Applications. 428(4), 992-1000.
Liu, S. and Neudecker, H. 1999. A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities, Statistical Papers. 40, 55–73.
