Abstract
The aim of this paper is to present the structure of a class of matrices that enables explicit inverse to be obtained. Starting from an already known class of matrices, we construct a Hadamard product that derives the class under consideration. The latter are defined by parameters, analytic expressions of which provide the elements of the lower Hessenberg form inverse. Recursion formulae of these expressions reduce the arithmetic operations in evaluating the inverse to .
Highlights
In 1, a class of matrices Kn aij with elements ⎧⎨1, i j, aij ⎩aj, i > j1.1 is treated
The so-constructed class is defined by 4n − 2 parameters, and its inverse has a lower Hessenberg analytic expression
We prove that the expressions 2.3 give the inverse matrix M−1
Summary
The class under consideration is defined by the Hadamard product of Gn and a matrix L, which results from Gn first by assigning the values. Advances in Numerical Analysis ai ln−i 1 and bi kn−i 1 to the latter in order to get a matrix K, say, and by the relation L P KT P , where P pij is the permutation matrix with elements. ⎨1, i n − j 1, pij ⎩0, otherwise. The so-constructed class is defined by 4n − 2 parameters, and its inverse has a lower Hessenberg analytic expression. It is worth noting that the classes L and Gn that produce the class M L ◦ Gn belong to the extended DIM classes presented in 3 as well as to the categories of the upper and lower Brownian matrices, respectively, as they have been defined in 4
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