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Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices

100%

Verde-Star L.

tom 3

nr 1

We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

A new method for solving ill-conditioned linear systems

88%

Soleymani F.

tom Vol. 33, no. 2

337--344

An accurate numerical method is established for matrix inversion. It is shown theoretically that the scheme possesses the high order of convergence of seven. Subsequently, the method is taken into account for solving linear systems of equations. The accuracy of the contributed iterative method is clarified on solving numerical examples when the coefficient matrices are ill-conditioned. All of the computations are performed on a PC using several programs written in MATHEMATICA 7.

Inversion of selected structures of block matrices of chosen mechatronic systems

63%

Trawiński T. , Kochan A. , Kielan P. , Kurzyk D.

tom Vol. 64, nr 4

853--863

This paper describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occurs, among others, in mathematical models of modern positioning systems of mass storage devices. The inversion method of block matrices is presented as well. The presented form of general formulas describing the calculation complexity of inverted form of block matrix were prepared for three different cases of division into internal blocks. The obtained results are compared with a standard Gaussian method and the “inv” method used in Matlab. The proposed method for matrix inversion is much more effective in comparison in standard Matlab matrix inversion “inv” function (almost two times faster) and is much less numerically complex than standard Gauss method.