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Advances in Numerical Analysis

VolumeΒ 2011Β (2011), Article IDΒ 593548, 6 pages

http://dx.doi.org/10.1155/2011/593548

## A Generalization of a Class of Matrices: Analytic Inverse and Determinant

Department of Mathematics, University of Patras, 26500 Patras, Greece

Received 29 December 2010; Revised 3 October 2011; Accepted 10 October 2011

Academic Editor: AlfredoΒ Bermudez De Castro

Copyright Β© 2011 F. N. Valvi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 .

#### 1. Introduction

In [1], a class of matrices with elementsis treated. A generalization of this class is presented in [2] by the matrix , where

In this paper, we consider a more extended class of matrices, , and we deduce in analytic form its inverse and determinant. The class under consideration is defined by the Hadamard product of and a matrix , which results from first by assigning the values and to the latter in order to get a matrix , say, and then by the relation , where is the permutation matrix with elements The so-constructed class is defined by parameters, and its inverse has a lower Hessenberg analytic expression. By assigning particular values to these parameters, a great variety of test matrices occur.

It is worth noting that the classes and that produce the class 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].

#### 2. The Class of Matrices and Its Inverse

Let be the matrix with elements that is, If is its inverse, then the following expressions give its elements where with and with the obvious assumption

#### 3. The Proof

We prove that the expressions (2.3) give the inverse matrix . To that purpose, we reduce to the identity matrix by applying elementary row operations. Then the product of the corresponding elementary matrices gives the inverse matrix. In particular, adopting the conventions (2.4), we apply the following sequence of row operations:

*Operation. *, which gives the lower triangular matrix

*Operation. *, which results in a bidiagonal matrix with main diagonal
and lower first diagonal

*Operation. *which gives the diagonal matrix

*Operation. *, which gives the identity matrix.

Operations 1β4 transform the identity matrix to the following forms, respectively:

*Form. *The upper bidiagonal matrix consisting of the main diagonal
and the upper first diagonal

*Form. *The tridiagonal matrix

*Form. *The lower Hessenberg matrix
where the symbol stands for the quantity .

*Form. *The matrix whose elements are given by the expressions (2.4).

The determinant of takes the form
Evidently, is singular if for some .

#### 4. Numerical Complexity

The inverse of the matrix is given explicitly by the expressions (2.3). However, a careful reader could easily derive the recursive algorithm that gives the elements under the main diagonal of . In particular,

or, alternatively,

where the , , , and are given by the relations (2.4). By use of the above algorithms, the estimation of the whole inverse of the matrix is carried out in multiplications/divisions, since the coefficient of depends only on the second (first) subscript, respectively, and in additions/subtractions.

#### 5. Remarks

When replacing , , and by , , and , respectively, the matrix (see (2.2)) is transformed into the transpose of the matrix [5, Section 2]. However, the primary fact for a test matrix is the structure of its particular pattern, which succeeds in yielding the analytic expression of its inverse. In the present case, the determinants of the minors of the elements , vanish to provide the lower Hessenberg type inverse. In detail, each minor of , that occurs after having removed the th row and th column, , has the determinant of its upper left minor equal to zero, since the last two columns of the latter are linearly dependent. Accordingly, by using induction, it can be proved that all the remaining upper left minors of order vanish.

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