Document Type: Research Paper

Authors

1 Department of Statistics, Athens University of Economics and Business, 76 Patission Str, 10434, Athens, Greece

2 Department of Economics, Division of Mathematics and Informatics, National and Kapodistrian University of Athens, Athens, Greece

3 Department of Computer Science, Faculty of Science and Mathematics, University of Nis, Visegradska 33, 18000 Nis, Serbia

Abstract

Following the results of \cite{Med}, regarding the Aluthge transform of polynomial matrices, the symbolic computation of the Duggal transform of a polynomial matrix $A$ is developed in this paper, using the polar decomposition and the singular value decomposition of $A$. Thereat, the polynomial singular value decomposition method is utilized, which is an iterative algorithm with numerical characteristics. The introduced algorithm is proven and illustrated in numerical examples. We also represent symbolically the Duggal transform of rank-one matrices using cross products of vectors and show that the Duggal transform of such matrices can be given explicitly by a closed formula and is equal to its Aluthge transform.

Keywords

Main Subjects

[1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integ. Equ. Oper. Theory. 13 (1990), 307-315.

[2] C. Foias, B. Jung, E. Ko, C. Pearcy, Complete contractivity of maps associated with the Aluthge and Duggal transforms. Paci c J. Math. 209 (2) (2003), 249-259.

[3] J. Foster, J. Chambers, J. McWhirter, A novel algorithm for calculating the QR decomposition of a polynomial matrix, IEEE Acoustics, Speech and Signal Processing, ICICS, 2009.

[4] J. Foster, J. McWhirter, M. Davies, An algorithm for calculating the QR and singular value decompositions of polynomial matrices, IEEE Transactions on Signal Processing. 58 (3) (2009), 1263-1274.

[5] J. Foster, Algorithms and techniques for polynomial matrix decompositions, Ph.D. dissertation, School Eng, Cardiff Univ, U.K., 2008.

[6] D. Pappas, V. N. Katsikis, I. P. Stanimirovic, Symbolic computation of the Aluthge transform. Mediterr. J. Math. (2017), doi:10.1007/s00009-017-0862-5.

[7] J. Ringrose, Compact non self adjoint operators, Van Nostrand London, 1971.

[8] R. Wirski, K. Wawryn, Decomposition of rational matrix functions, Information. Communications and Signal Processing, ICICS, 2009.

[9] J. G. McWhirter, An algorithm for polynomial matrix SVD based on generalized Kogbetliantz transformations, Proceedings of EUSIPCO, 2010.