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Advances in Numerical Analysis
Volume 2011 (2011), Article ID 593548, 6 pages
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: Alfredo Bermudez De Castro
Copyright © 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.
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 .
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  as well as to the categories of the upper and lower Brownian matrices, respectively, as they have been defined in .
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.
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,
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.
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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