Some Contributions to the Theory of Matrices and Coupled Matrices Rhotrices

Abstract

The present thesis is concerned with some aspects of matrices and rhotrices. The newlineaim of the matrix investigation is to obtain the formula for trace of powers of a newlinesquare matrix A belongs Mm. This is achieved by first investigating two particular orders of matrices, viz. 2 x 2 and 3 x 3 matrices, followed by framing and proving the formula newlinefor Tr(An) for A belongs Mm. Switching over to the investigation of the element sum of newlinepowers of a matrix, again the matrices of order 2 and 3 are analysed and formulae for newlinesu(An), (A belongs M2 or A belongs M3), are obtained. The intrinsic formulae for sum for higher order is extremely involved and lengthy, restricting us to conjecture the formulae mfor su(An) for (A belongs M4 or A belongs M5). Of course the the conjectures are based upon verification of large number of typical matrices on MatLab. The rhotrix, that is, couple of a t x t matrix with a (t - 1) x (t - 1) matrix, is comparatively a new concept defined by [1] in 2003. It still laked the analysis of results analogous to their counter parts for matrices. We do so. Prominent part of this is with the matrix multiplication and some part is with the Hadamard product. Then we divert to the algebraic structures of rhotrices over a commutative unital ring as well as over Z and C. This part is fully with a new product called heart oriented product. newline