Generalized Inversions of Hadamard and Tensor Products for Matrices
Author(s)
Saburou Saitoh*
ABSTRACT
We shall give natural generalized solutions of Hadamard and tensor products equations for matrices by the concept of the Tikhonov regularization combined with the theory of reproducing kernels.
KEYWORDS
Reproducing Kernel, Positive Definite Hermitian Matrix, Tensor Product, Hadamard Product, Generalized Inverse, Matrix Equation, Tikhonov Regularization, 100/0 = 0, 0/0 = 0, Generalized Fraction, Generalized Fractional Function
Reproducing Kernel, Positive Definite Hermitian Matrix, Tensor Product, Hadamard Product, Generalized Inverse, Matrix Equation, Tikhonov Regularization, 100/0 = 0, 0/0 = 0, Generalized Fraction, Generalized Fractional Function
Cite this paper
Saitoh, S. (2014) Generalized Inversions of Hadamard and Tensor Products for Matrices. Advances in Linear Algebra & Matrix Theory, 4, 87-95. doi: 10.4236/alamt.2014.42006.
Saitoh, S. (2014) Generalized Inversions of Hadamard and Tensor Products for Matrices. Advances in Linear Algebra & Matrix Theory, 4, 87-95. doi: 10.4236/alamt.2014.42006.
References
[1] Asaduzzaman, M. and Saitoh, S. (2003) Inverses of a Family of Matrices and Generalizations of Pythagorean Theorem. Panamerican Mathematical Journal, 13, 45-53. [2] Mai, W., Yan, M., Qian, T., Riva, M.D. and Saitoh, S. (1013) A Matrix Inequality for the Inversions of the Restrictions of a Positive Definite Hermitian Matrix. Advances in Linear Algebra & Matrix Theory, 3, 55-58. [3] Mond, B., Pecaric, J.E. and Saitoh, S. (1994) History, Variations and Generalizations of an Inequality of Marcus. Riazi: Journal of Karachi Mathematical Association, 16, 7-15. [4] Saitoh, S. (1982) Positive Definite Hermitian Matrices and Reproducing Kernels. Linear Algebra and Its Applications, 48, 119-130.
http://dx.doi.org/10.1016/0024-3795(82)90102-1 [5] Saitoh, S. (1987) Quadratic Inequalities Deduced From the Theory of Reproducing Kernels. Linear Algebra and Its Applications, 93, 171-178. http://dx.doi.org/10.1016/S0024-3795(87)90322-3 [6] Saitoh, S. (1988) Quadratic Inequalities Associated with Integrals of Reproducing Kernels. Linear Algebra and Its Applications, 101, 269-280.
http://dx.doi.org/10.1016/0024-3795(88)90154-1 [7] Saitoh, S. (2003) Generalizations of the Triangle Inequality. Journal of Inequalities in Pure and Applied Mathematics, 4, Article 62. [8] Sawano, Y. (2011) Pasting Reproducing Kernel Hilbert Spaces. Jaen Journal on Approximation, 3, 135-141. [9] Yamada, A. (2012) Oppenheim’s Inequality and RKHS. Mathematical Inequalities and Applications, 15, 449-456. [10] Saitoh, S. (1997) Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Math. Series 369, Addison Wesley Longman, Harlow. [11] Saitoh, S. (2010) Theory of Reproducing Kernels: Applications to Approximate Solutions of Bounded Linear Operator Functions on Hilbert Spaces. American Mathematical Society Translations: Series 2, 230, Amer. Math. Soc., Providence, RI. [12] Castro, L.P., Fujiwara, H., Saitoh, S., Sawano, Y., Yamada, A. and Yamada, M. (2012) Fundamental Error Estimates Inequalities for the Tikhonov Regularization Using Rreproducing Kernels. International Series of Numerical Mathematics 161, Inequalities and Appications 2010, Springer, Basel, 87-101. [13] Castro, L.P., Saitoh, S., Sawano, Y. and Silva, A.S. (2012) Discrete Linear Differential Equations. Analysis, 32, 181- 191.
http://dx.doi.org/10.1524/anly.2012.1104 [14] Castro, L.P., Fujiwara, H., Rodrigues, M. M., Saitoh, S. and Tuan, V.K. (2014) Aveiro Discretization Method in Mathematics: A New Discretization Principle. In: Pardalos, P. and Rassias, T.M., Eds., Mathematics without Boundaries: Surveys in Pure Mathematics, in press. [15] Castro, L.P., Fujiwara, H., Qian, T. and Saitoh, S. (2014) How to Catch Smoothing Properties and Analyticity of Functions by Computers? In: Pardalos, P. and Rassias, T.M., Eds., Mathematics without Boundaries: Surveys in Interdisipinary Research, in press. [16] Castro, L. P., Saitoh, S. and Yamada, A. (2014) Representations of Solutions of Tichhonov Functional Equations and Applications to Multiplication Operators of the Szego Spaces (submitted). [17] Takahashi, S. (2014) On the Identities 100/0=0 and 0/0=0. (submitted). [18] Kuroda, T., Michiwaki, H., Saitoh, S. and Yamane, M. (2014) New Meanings of the Division by Zero and Interpretations on 100/0 = 0 and on 0/0 = 0. International Journal of Applied Mathematics, 27, 191-198.
[1] Asaduzzaman, M. and Saitoh, S. (2003) Inverses of a Family of Matrices and Generalizations of Pythagorean Theorem. Panamerican Mathematical Journal, 13, 45-53. [2] Mai, W., Yan, M., Qian, T., Riva, M.D. and Saitoh, S. (1013) A Matrix Inequality for the Inversions of the Restrictions of a Positive Definite Hermitian Matrix. Advances in Linear Algebra & Matrix Theory, 3, 55-58. [3] Mond, B., Pecaric, J.E. and Saitoh, S. (1994) History, Variations and Generalizations of an Inequality of Marcus. Riazi: Journal of Karachi Mathematical Association, 16, 7-15. [4] Saitoh, S. (1982) Positive Definite Hermitian Matrices and Reproducing Kernels. Linear Algebra and Its Applications, 48, 119-130.
http://dx.doi.org/10.1016/0024-3795(82)90102-1 [5] Saitoh, S. (1987) Quadratic Inequalities Deduced From the Theory of Reproducing Kernels. Linear Algebra and Its Applications, 93, 171-178. http://dx.doi.org/10.1016/S0024-3795(87)90322-3 [6] Saitoh, S. (1988) Quadratic Inequalities Associated with Integrals of Reproducing Kernels. Linear Algebra and Its Applications, 101, 269-280.
http://dx.doi.org/10.1016/0024-3795(88)90154-1 [7] Saitoh, S. (2003) Generalizations of the Triangle Inequality. Journal of Inequalities in Pure and Applied Mathematics, 4, Article 62. [8] Sawano, Y. (2011) Pasting Reproducing Kernel Hilbert Spaces. Jaen Journal on Approximation, 3, 135-141. [9] Yamada, A. (2012) Oppenheim’s Inequality and RKHS. Mathematical Inequalities and Applications, 15, 449-456. [10] Saitoh, S. (1997) Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Math. Series 369, Addison Wesley Longman, Harlow. [11] Saitoh, S. (2010) Theory of Reproducing Kernels: Applications to Approximate Solutions of Bounded Linear Operator Functions on Hilbert Spaces. American Mathematical Society Translations: Series 2, 230, Amer. Math. Soc., Providence, RI. [12] Castro, L.P., Fujiwara, H., Saitoh, S., Sawano, Y., Yamada, A. and Yamada, M. (2012) Fundamental Error Estimates Inequalities for the Tikhonov Regularization Using Rreproducing Kernels. International Series of Numerical Mathematics 161, Inequalities and Appications 2010, Springer, Basel, 87-101. [13] Castro, L.P., Saitoh, S., Sawano, Y. and Silva, A.S. (2012) Discrete Linear Differential Equations. Analysis, 32, 181- 191.
http://dx.doi.org/10.1524/anly.2012.1104 [14] Castro, L.P., Fujiwara, H., Rodrigues, M. M., Saitoh, S. and Tuan, V.K. (2014) Aveiro Discretization Method in Mathematics: A New Discretization Principle. In: Pardalos, P. and Rassias, T.M., Eds., Mathematics without Boundaries: Surveys in Pure Mathematics, in press. [15] Castro, L.P., Fujiwara, H., Qian, T. and Saitoh, S. (2014) How to Catch Smoothing Properties and Analyticity of Functions by Computers? In: Pardalos, P. and Rassias, T.M., Eds., Mathematics without Boundaries: Surveys in Interdisipinary Research, in press. [16] Castro, L. P., Saitoh, S. and Yamada, A. (2014) Representations of Solutions of Tichhonov Functional Equations and Applications to Multiplication Operators of the Szego Spaces (submitted). [17] Takahashi, S. (2014) On the Identities 100/0=0 and 0/0=0. (submitted). [18] Kuroda, T., Michiwaki, H., Saitoh, S. and Yamane, M. (2014) New Meanings of the Division by Zero and Interpretations on 100/0 = 0 and on 0/0 = 0. International Journal of Applied Mathematics, 27, 191-198.