Explicit Inversion for Two Brownian-Type Matrices

Florendia Valvi, Vassilis Geroyannis

Department of Mathematics, University of Patras, Patras, Greece.
Department of Physics, University of Patras, Patras, Greece.

DOI: 10.4236/am.2012.39157                

Abstract

We present explicit inverses of two Brownian-type matrices, which are defined as Hadamard products of certain already known matrices. The matrices under consideration are defined by 3n - 1 parameters and their lower Hessenberg form inverses are expressed analytically in terms of these parameters. Such matrices are useful in the theory of digital signal processing and in testing matrix inversion algorithms.

Keywords

Brownian Matrix; Hadamard Product; Hessenberg Matrix; Numerical Complexity; Test Matrix

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. J. Herbold, “A Generalization of a Class of Test Matrices,” Mathematics of Computation, Vol. 23, 1969, pp. 823-826. doi:10.1090/S0025-5718-1969-0258259-0
[2]	F. N. Valvi, “Explicit Presentation of the Inverses of Some Types of Matrices,” IMA Journal of Applied Mathematics, Vol. 19, No. 1, 1977, pp. 107-117. doi:10.1093/imamat/19.1.107
[3]	M. J. C. Gover and S. Barnett, “Brownian Matrices: Properties and Extensions,” International Journal of Systems Science, Vol. 17, No. 2, 1986, pp. 381-386. doi:10.1080/00207728608926813
[4]	B. Picinbono, “Fast Algorithms for Brownian Matrices,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 31, No. 2, 1983, pp. 512-514. doi:10.1109/TASSP.1983.1164078
[5]	G. Carayannis, N. Kalouptsidis and D. G. Manolakis, “Fast Recursive Algorithms for a Class of Linear Equations,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 30, No. 2, 1982, pp. 227-239. doi:10.1109/TASSP.1982.1163876
[6]	H. W. Milnes, “A Note Concerning the Properties of a Certain Class of Test Matrices,” Mathematics of Computation, Vol. 22, 1968, pp. 827-832. doi:10.1090/S0025-5718-1968-0239743-1
[7]	R. T. Gregory and D. L. Karney, “A Collection of Matrices for Testing Computational Algorithms,” Wiley-Interscience, London, 1969.