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On Condition Numbers for the Weighted Moore-Penrose Inverse and the Weighted Least Squares Problem involving Kronecker Products

Published online by Cambridge University Press:  28 May 2015

T. T. Chen  and
W. Li
T. T. Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
W. Li
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
*
Corresponding author. Email: liwen@scnu.edu.cn
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Abstract

We establish some explicit expressions for norm-wise, mixed and componentwise condition numbers for the weighted Moore-Penrose inverse of a matrix A ⊗ B and more general matrix function compositions involving Kronecker products. The condition number for the weighted least squares problem (WLS) involving a Kronecker product is also discussed.

Keywords

65F10(Weighted) Moore-Penrose inverseweighted least squaresKronecker productcondition number
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 1 , February 2014 , pp. 1 - 20
Copyright
Copyright © Global-Science Press 2014

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References

[1]Arioli, M., Baboulin, M. and Gratton, S., A partial condition number for linear least squares problems, SIAM J. Matrix Anal. Appl. 29, 413–433 (2007).CrossRefGoogle Scholar
[2]Baboulin, M., Dongarra, J., Gratton, S. and Langou, J., Computing the conditioning of the components of a linear least-squares solution, Numer. Linear Algebra Appl. 16, 517–533 (2009).Google Scholar
[3]Baboulin, M. and Gratton, S., Using dual techniques to derive component-wise and mixed condition numbers for a linear function of a linear least squares solution, BIT 49, 3–19 (2009).CrossRefGoogle Scholar
[4]Ben-Israel, A. and EGreville, T.N., Generalized Inverses: Theory and Applications, 2nd Edition, Springer Verlag: New York, 2003.Google Scholar
[5]Björck, Å., Component-wise perturbation analysis and error bounds for linear least squares solutions, BIT 31, 238–244 (1991).Google Scholar
[6]Björck, Å., Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1996.CrossRefGoogle Scholar
[7]Brewer, J., Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits Syst. 25, 772–781 (1978).CrossRefGoogle Scholar
[8]Brewer, J., Correction to ‘Kronecker products and matrix calculus in system theory’, IEEE Trans. Circuits Syst. 26, 360–360 (1979).Google Scholar
[9]Canta, G.R. and Poggi, G., Multispectral-image coding by vector quantization with Kronecker product representation, Proceedings of the International Conference on Image Processing, vol. 2, pp. 931–934, Lausanne, Switzerland, September 1996.Google Scholar
[10]Diao, H., Wang, W., Wei, Y. and Qiao, S., On condition numbers for Moore-Penrose inverse and linear least squates problem involving Kronecker products, Numer. Linear Algebra Appl. 20, 44–59 (2013).Google Scholar
[11]Gohberg, I. and Koltracht, I., Mixed, component-wise, and structured condition numbers, SIAM J. Matrix Anal. Appl. 14, 688–704 (1993).Google Scholar
[12]Golub, G.H. and Van Loan, C.F., Matrix Computations, 3rd Edition, Johns Hopkins University: Baltimore, 1996.Google Scholar
[13]Graham, A., Kronecker Products and MatrixCalculus with Applixations, Ellis Howood Ltd: New York, USA, 1981.Google Scholar
[14]Higham, N.J., Perturbation theory and backward error for AX - XB = C, BIT 33, 124–136 (1993).Google Scholar
[15]Kressner, D. and Tobler, C., Krylov subspace methods for linear systems with Tensor product structure, SIAM J. Matrix Anal. Appl. 31, 1688–1714 (2010).CrossRefGoogle Scholar
[16]Langville, A.N. and Stewart, W.J., A Kronecker product approximate preconditioner for SANs, Numer. Linear Algebra Appl. 11, 723–752 (2004).Google Scholar
[17]Li, Z. and Sun, J., Mixed and component-wise condition numbers for weighted Moore-Penrose inverse and weighted least squares problems, Filomat 23, 43–59 (2009).Google Scholar
[18]Lin, L., Lu, T. and Wei, Y., On level-2 condition number for the weighted Moore-Penrose inverse, Comput. Math. Appl. 55, 788–800 (2008).Google Scholar
[19]Magnus, J.R. and Neudecker, H., Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition, John Wiley Sons Inc.: Hoboken, NJ, USA, 2007.Google Scholar
[20]Rice, J.R., A theory of condition, SIAM J. Numer. Anal. 3, 217–232 (1966).Google Scholar
[21]Rohn, J., New condition numbers for matrices and linear systems, Computing 41, 167–169 (1989).Google Scholar
[22]Schott, J.R., Matrix Analysis for Statistics, 2nd Edition, John Wiley Sons Inc.: Hoboken, NJ, USA, 2005.Google Scholar
[23]Suehiro, N., Chenggao, H. and Imoto, T., Very efficient wireless frequency usage based on pseudocoherent addition of multipath signals using kronecker product with rows of dft matrix, Proceedings of the IEEE International Symposium on Information Theory, vol. 29, pp. 385–385, Yokohama, Japan, 2003.Google Scholar
[24]Wang, G., Weighted Moore-Penrose, Drazin, and group inverses of the Kronecker product A ⊗ B, and some applications, Linear Algerbra Appl. 250, 39–50 (1997).Google Scholar
[25]Wei, M., Supremum and Stability of Weighted Pseudoinverses and Weighted Least Squares Problems, Analysis and Computations, Nova Science Pubishers: Huntington, NY, 2001.Google Scholar
[26]Wu, C.W. and Chua, L.O., Application of Kronecker products to the analysis of systems with uniform linear couping, IEEE Trans. Circuits Syst. 42, 775–778 (1995).Google Scholar
[27]Xiang, H., Diao, H. and Wei, Y., On perturbation bounds of Kronecker product linear systems and their level-2 condition numbers, J. Comput. Appl. Math. 183, 210–231 (2005).Google Scholar