About this Journal Submit a Manuscript Table of Contents 
Advances in Numerical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 593548, 6 pages
doi:10.1155/2011/593548
Research Article

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

F. N. Valvi

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 4 𝑛 βˆ’ 2 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 π’ͺ ( 𝑛 2 ) .

1. Introduction

In [1], a class of matrices 𝐾 𝑛 = [ π‘Ž 𝑖 𝑗 ] with elements π‘Ž 𝑖 𝑗 = ξƒ― π‘Ž 1 , 𝑖 β©½ 𝑗 , 𝑗 , 𝑖 > 𝑗 ( 1 . 1 ) is treated. A generalization of this class is presented in [2] by the matrix 𝐺 𝑛 = [ 𝑏 𝑖 𝑗 ] , where 𝑏 𝑖 𝑗 = ξƒ― 𝑏 𝑗 π‘Ž , 𝑖 β©½ 𝑗 , 𝑗 , 𝑖 > 𝑗 . ( 1 . 2 )

In this paper, we consider a more extended class of matrices, 𝑀 , and we deduce in analytic form its inverse and determinant. The class under consideration is defined by the Hadamard product of 𝐺 𝑛 and a matrix 𝐿 , which results from 𝐺 𝑛 first by assigning the values π‘Ž 𝑖 = 𝑙 𝑛 βˆ’ 𝑖 + 1 and 𝑏 𝑖 = π‘˜ 𝑛 βˆ’ 𝑖 + 1 to the latter in order to get a matrix 𝐾 , say, and then by the relation 𝐿 = 𝑃 𝐾 𝑇 𝑃 , where 𝑃 = [ 𝑝 𝑖 𝑗 ] is the permutation matrix with elements 𝑝 𝑖 𝑗 = ξƒ― 1 , 𝑖 = 𝑛 βˆ’ 𝑗 + 1 , 0 , o t h e r w i s e . ( 1 . 3 ) The so-constructed class is defined by 4 𝑛 βˆ’ 2 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 𝐿 and 𝐺 𝑛 that produce the class 𝑀 = 𝐿 ∘ 𝐺 𝑛 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 𝑀 = [ π‘š 𝑖 𝑗 ] be the matrix with elements π‘š 𝑖 𝑗 = ξƒ― π‘˜ 𝑖 𝑏 𝑗 𝑙 , 𝑖 β©½ 𝑗 , 𝑖 π‘Ž 𝑗 , 𝑖 > 𝑗 , ( 2 . 1 ) that is, ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ π‘˜ 𝑀 = 1 𝑏 1 π‘˜ 1 𝑏 2 π‘˜ 1 𝑏 3 β‹― π‘˜ 1 𝑏 𝑛 βˆ’ 1 π‘˜ 1 𝑏 𝑛 𝑙 2 π‘Ž 1 π‘˜ 2 𝑏 2 π‘˜ 2 𝑏 3 β‹― π‘˜ 2 𝑏 𝑛 βˆ’ 1 π‘˜ 2 𝑏 𝑛 𝑙 3 π‘Ž 1 𝑙 3 π‘Ž 2 π‘˜ 3 𝑏 3 β‹― π‘˜ 3 𝑏 𝑛 βˆ’ 1 π‘˜ 3 𝑏 𝑛 β‹― 𝑙 𝑛 βˆ’ 1 π‘Ž 1 𝑙 𝑛 βˆ’ 1 π‘Ž 2 𝑙 𝑛 βˆ’ 1 π‘Ž 3 β‹― π‘˜ 𝑛 βˆ’ 1 𝑏 𝑛 βˆ’ 1 π‘˜ 𝑛 βˆ’ 1 𝑏 𝑛 𝑙 𝑛 π‘Ž 1 𝑙 𝑛 π‘Ž 2 𝑙 𝑛 π‘Ž 3 β‹― 𝑙 𝑛 π‘Ž 𝑛 βˆ’ 1 π‘˜ 𝑛 𝑏 𝑛 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ . ( 2 . 2 ) If 𝑀 βˆ’ 1 = [ πœ‡ 𝑖 𝑗 ] is its inverse, then the following expressions give its elements πœ‡ 𝑖 𝑗 = ⎧ βŽͺ βŽͺ βŽͺ βŽͺ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ βŽͺ βŽͺ βŽͺ βŽͺ ⎩ π‘˜ 𝑖 + 1 𝑏 𝑖 βˆ’ 1 βˆ’ 𝑙 𝑖 + 1 π‘Ž 𝑖 βˆ’ 1 𝑐 𝑖 βˆ’ 1 𝑐 𝑖 π‘˜ , 𝑖 = 𝑗 = 2 , 3 , … , 𝑛 βˆ’ 1 , 2 𝑐 0 𝑐 1 𝑏 , 𝑖 = 𝑗 = 1 , 𝑛 βˆ’ 1 𝑐 𝑛 βˆ’ 1 𝑐 𝑛 ( , 𝑖 = 𝑗 = 𝑛 , βˆ’ 1 ) 𝑖 + 𝑗 𝑑 𝑗 βˆ’ 1 𝑔 𝑖 ∏ 𝑖 βˆ’ 1 𝜈 = 𝑗 + 1 𝑓 𝜈 ∏ 𝑖 𝜈 = 𝑗 βˆ’ 1 𝑐 𝜈 βˆ’ 1 , 𝑖 > 𝑗 , 𝑐 𝑖 , 𝑖 = 𝑗 βˆ’ 1 , 0 , 𝑖 < 𝑗 βˆ’ 1 , ( 2 . 3 ) where 𝑐 𝑖 = π‘˜ 𝑖 + 1 𝑏 𝑖 βˆ’ 𝑙 𝑖 + 1 π‘Ž 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 βˆ’ 1 , 𝑐 0 = π‘˜ 1 , 𝑐 𝑛 = 𝑏 𝑛 , 𝑑 𝑖 = π‘Ž 𝑖 + 1 𝑏 𝑖 βˆ’ π‘Ž 𝑖 𝑏 𝑖 + 1 , 𝑖 = 1 , 2 , … , 𝑛 βˆ’ 2 , 𝑑 0 = π‘Ž 1 , 𝑓 𝑖 = 𝑙 𝑖 π‘Ž 𝑖 βˆ’ π‘˜ 𝑖 𝑏 𝑖 𝑔 , 𝑖 = 2 , 3 , … , 𝑛 βˆ’ 1 , 𝑖 = π‘˜ 𝑖 + 1 𝑙 𝑖 βˆ’ π‘˜ 𝑖 𝑙 𝑖 + 1 , 𝑖 = 2 , 3 , … , 𝑛 βˆ’ 1 , 𝑔 𝑛 = 𝑙 𝑛 , ( 2 . 4 ) with 𝑖 βˆ’ 1  𝜈 = 𝑗 + 1 𝑓 𝜈 = 1 w h e n e v e r 𝑖 = 𝑗 + 1 , ( 2 . 5 ) and with the obvious assumption 𝑐 𝑖 β‰  0 , 𝑖 = 0 , 1 , 2 , … , 𝑛 . ( 2 . 6 )

3. The Proof

We prove that the expressions (2.3) give the inverse matrix 𝑀 βˆ’ 1 . To that purpose, we reduce 𝑀 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. r o w 𝑖 βˆ’ ( π‘˜ 𝑖 / π‘˜ 𝑖 + 1 ) Γ— r o w ( 𝑖 + 1 ) , 𝑖 = 1 , 2 , … , 𝑛 βˆ’ 1 , which gives the lower triangular matrix ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ π‘˜ 1 π‘˜ 2 𝑐 1 π‘Ž 0 β‹― 0 0 1 π‘˜ 3 𝑔 2 π‘˜ 2 π‘˜ 3 𝑐 2 π‘Ž β‹― 0 0 β‹― β‹― β‹― β‹― β‹― 1 π‘˜ 𝑛 𝑔 𝑛 βˆ’ 1 π‘Ž 2 π‘˜ 𝑛 𝑔 𝑛 βˆ’ 1 β‹― π‘˜ 𝑛 βˆ’ 1 π‘˜ 𝑛 𝑐 𝑛 βˆ’ 1 0 𝑔 𝑛 π‘Ž 1 𝑔 𝑛 π‘Ž 2 β‹― 𝑔 𝑛 π‘Ž 𝑛 βˆ’ 1 𝑔 𝑛 𝑏 𝑛 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ . ( 3 . 1 )

Operation. r o w 𝑖 βˆ’ ( π‘˜ 𝑖 𝑔 𝑖 / π‘˜ 𝑖 + 1 𝑔 𝑖 βˆ’ 1 ) Γ— r o w ( 𝑖 βˆ’ 1 ) , 𝑖 = 𝑛 , 𝑛 βˆ’ 1 , … , 3 , π‘˜ 𝑛 + 1 = 1 , which results in a bidiagonal matrix with main diagonal ξ‚΅ π‘˜ 1 𝑐 1 π‘˜ 2 , π‘˜ 2 𝑐 2 π‘˜ 3 π‘˜ , … , 𝑛 βˆ’ 1 𝑐 𝑛 βˆ’ 1 π‘˜ 𝑛 , π‘˜ 𝑛 𝑏 𝑛 ξ‚Ά ( 3 . 2 ) and lower first diagonal ξ‚΅ π‘Ž 1 𝑔 2 π‘˜ 3 , π‘˜ 3 𝑔 3 𝑓 2 π‘˜ 4 𝑔 2 π‘˜ , … , 𝑛 βˆ’ 1 𝑔 𝑛 βˆ’ 1 𝑓 𝑛 βˆ’ 2 π‘˜ 𝑛 𝑔 𝑛 βˆ’ 2 , π‘˜ 𝑛 𝑔 𝑛 𝑓 𝑛 βˆ’ 1 g 𝑛 βˆ’ 1 ξ‚Ά . ( 3 . 3 )

Operation. r o w 2 βˆ’ ( π‘˜ 2 π‘Ž 1 𝑔 2 / π‘˜ 1 π‘˜ 3 𝑐 1 ) Γ— r o w 1 , a n d r o w 𝑖 βˆ’ ( π‘˜ 𝑖 π‘˜ 𝑖 𝑔 𝑖 𝑓 𝑖 βˆ’ 1 / π‘˜ 𝑖 βˆ’ 1 π‘˜ 𝑖 + 1 𝑔 𝑖 βˆ’ 1 𝑐 𝑖 βˆ’ 1 ) Γ— r o w ( 𝑖 βˆ’ 1 ) , 𝑖 = 3 , 4 , … , 𝑛 , which gives the diagonal matrix  π‘˜ 1 𝑐 1 π‘˜ 2 π‘˜ 2 𝑐 2 π‘˜ 3 β‹― π‘˜ 𝑛 βˆ’ 1 𝑐 𝑛 βˆ’ 1 π‘˜ 𝑛 π‘˜ 𝑛 𝑏 𝑛  . ( 3 . 4 )

Operation. π‘˜ 𝑖 + 1 / π‘˜ 𝑖 𝑐 𝑖 Γ— r o w 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 , which gives the identity matrix.

Operations 1–4 transform the identity matrix to the following forms, respectively:

Form. The upper bidiagonal matrix consisting of the main diagonal ( 1 , 1 , … , 1 ) ( 3 . 5 ) and the upper first diagonal ξ‚΅ βˆ’ π‘˜ 1 π‘˜ 2 π‘˜ , βˆ’ 2 π‘˜ 3 π‘˜ , … , βˆ’ 𝑛 βˆ’ 1 π‘˜ 𝑛 ξ‚Ά . ( 3 . 6 )

Form. The tridiagonal matrix ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ π‘˜ 1 βˆ’ 1 π‘˜ 2 π‘˜ 0 β‹― 0 0 0 1 βˆ’ 2 π‘˜ 3 π‘˜ β‹― 0 0 0 βˆ’ 3 𝑔 3 π‘˜ 4 𝑔 2 π‘˜ 3 ξ€· π‘˜ 4 𝑙 2 βˆ’ π‘˜ 2 𝑙 4 ξ€Έ π‘˜ 4 𝑔 2 β‹― π‘˜ β‹― 0 0 0 0 0 β‹― 𝑛 βˆ’ 1 ξ€· π‘˜ 𝑛 𝑙 𝑛 βˆ’ 2 βˆ’ π‘˜ 𝑛 βˆ’ 2 𝑙 𝑛 ξ€Έ π‘˜ 𝑛 𝑔 𝑛 βˆ’ 2 βˆ’ π‘˜ 𝑛 βˆ’ 1 π‘˜ 𝑛 π‘˜ 0 0 0 β‹― βˆ’ 𝑛 𝑔 𝑛 𝑔 𝑛 βˆ’ 1 π‘˜ 𝑛 𝑙 𝑛 βˆ’ 1 𝑔 𝑛 βˆ’ 1 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ . ( 3 . 7 )

Form. The lower Hessenberg matrix ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ π‘˜ 1 βˆ’ 1 π‘˜ 2 βˆ’ π‘Ž β‹― 0 0 1 𝑔 2 π‘˜ 2 π‘˜ 1 π‘˜ 3 𝑐 1 π‘˜ 2 π‘˜ 3 𝑐 1 ξ€· π‘˜ 3 𝑏 1 βˆ’ 𝑙 3 π‘Ž 1 ξ€Έ π‘Ž β‹― 0 0 1 𝑔 3 π‘˜ 3 𝑓 2 π‘˜ 1 π‘˜ 4 𝑐 1 𝑐 2 βˆ’ 𝑑 1 𝑔 3 π‘˜ 3 π‘˜ 4 𝑐 1 𝑐 2 β‹― 𝑠 π‘Ž β‹― 0 0 1 𝑔 𝑛 βˆ’ 1 π‘˜ 𝑛 βˆ’ 1 𝑓 2 β‹― 𝑓 𝑛 βˆ’ 2 π‘˜ 1 π‘˜ 𝑛 𝑐 1 β‹― 𝑐 𝑛 βˆ’ 2 𝑠 𝑑 1 𝑔 𝑛 βˆ’ 1 π‘˜ 𝑛 βˆ’ 1 𝑓 3 β‹― 𝑓 𝑛 βˆ’ 2 π‘˜ 𝑛 𝑐 1 𝑐 2 β‹― 𝑐 𝑛 βˆ’ 2 β‹― π‘˜ 𝑛 βˆ’ 1 k 𝑛 𝑐 𝑛 βˆ’ 2 ξ€· π‘˜ 𝑛 𝑏 𝑛 βˆ’ 2 βˆ’ 𝑙 𝑛 π‘Ž 𝑛 βˆ’ 2 ξ€Έ βˆ’ π‘˜ 𝑛 βˆ’ 1 π‘˜ 𝑛 𝑠 π‘Ž 1 𝑔 𝑛 π‘˜ 𝑛 𝑓 2 β‹― 𝑓 𝑛 βˆ’ 1 π‘˜ 1 𝑐 1 β‹― 𝑐 𝑛 βˆ’ 1 𝑠 𝑑 1 𝑔 𝑛 π‘˜ 𝑛 𝑓 3 β‹― 𝑓 𝑛 βˆ’ 1 𝑐 1 𝑐 2 β‹― 𝑐 𝑛 βˆ’ 1 𝑑 β‹― βˆ’ 𝑛 βˆ’ 2 𝑔 𝑛 π‘˜ 𝑛 𝑐 𝑛 βˆ’ 2 𝑐 𝑛 βˆ’ 1 𝑏 𝑛 βˆ’ 1 π‘˜ 𝑛 𝑐 𝑛 βˆ’ 1 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ , ( 3 . 8 ) where the symbol 𝑠 stands for the quantity ( βˆ’ 1 ) 𝑖 + 𝑗 .

Form. The matrix whose elements are given by the expressions (2.4).
The determinant of 𝑀 takes the form d e t ( 𝑀 ) = π‘˜ 1 𝑏 𝑛 ξ€· π‘˜ 2 𝑏 1 βˆ’ 𝑙 2 π‘Ž 1 ξ€Έ β‹― ξ€· π‘˜ 𝑛 𝑏 𝑛 βˆ’ 1 βˆ’ 𝑙 𝑛 π‘Ž 𝑛 βˆ’ 1 ξ€Έ . ( 3 . 9 ) Evidently, 𝑀 is singular if 𝑐 𝑖 = 0 for some 𝑖 ∈ { 0 , 1 , 2 , … , 𝑛 } .

4. Numerical Complexity

The inverse of the matrix 𝑀 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 𝑀 βˆ’ 1 . In particular, πœ‡ 𝑖 , 𝑖 βˆ’ 1 𝑑 = βˆ’ 𝑖 βˆ’ 2 𝑔 𝑖 𝑐 𝑖 βˆ’ 2 𝑐 𝑖 βˆ’ 1 𝑐 𝑖 πœ‡ , 𝑖 = 2 , 3 , … , 𝑛 , 𝑖 , 𝑖 βˆ’ 𝑠 βˆ’ 1 𝑑 = βˆ’ 𝑖 βˆ’ 𝑠 βˆ’ 2 𝑓 𝑖 βˆ’ 𝑠 𝑑 𝑖 βˆ’ 𝑠 βˆ’ 1 𝑐 𝑖 βˆ’ 𝑠 βˆ’ 2 πœ‡ 𝑖 , 𝑖 βˆ’ 𝑠 , 𝑖 = 3 , 4 , … , 𝑛 , 𝑠 = 1 , 2 , … , 𝑖 βˆ’ 2 , ( 4 . 1 )

or, alternatively, πœ‡ 𝑗 + 1 , 𝑗 𝑑 = βˆ’ 𝑗 βˆ’ 1 𝑔 𝑗 + 1 𝑐 𝑗 βˆ’ 1 𝑐 𝑗 𝑐 𝑗 + 1 πœ‡ , 𝑗 = 1 , 2 , … , 𝑛 βˆ’ 1 , 𝑗 + 𝑠 + 1 , 𝑗 𝑔 = βˆ’ 𝑗 + 𝑠 + 1 𝑓 𝑗 + 𝑠 𝑔 𝑗 + 𝑠 𝑐 𝑗 + 𝑠 + 1 πœ‡ 𝑗 + 𝑠 , 𝑗 , 𝑗 = 1 , 2 , … , 𝑛 βˆ’ 2 , 𝑠 = 1 , 2 , … , 𝑛 βˆ’ 𝑗 βˆ’ 1 , ( 4 . 2 )

where the 𝑐 𝑖 , 𝑑 𝑖 , 𝑓 𝑖 , and 𝑔 𝑖 are given by the relations (2.4). By use of the above algorithms, the estimation of the whole inverse of the matrix 𝑀 is carried out in 2 𝑛 2 + 1 1 𝑛 βˆ’ 1 9 multiplications/divisions, since the coefficient of πœ‡ 𝑖 𝑗 depends only on the second (first) subscript, respectively, and in 5 𝑛 βˆ’ 9 additions/subtractions.

5. Remarks

When replacing π‘˜ 𝑖 , 𝑏 𝑖 , and π‘Ž 𝑖 by π‘Ž 𝑖 , π‘˜ 𝑖 , and 𝑏 𝑖 , respectively, the matrix 𝑀 (see (2.2)) is transformed into the transpose 𝐢 𝑇 of the matrix 𝐢 [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 π‘š 𝑖 𝑗 , 𝑖 > 𝑗 + 1 , vanish to provide the lower Hessenberg type inverse. In detail, each minor of 𝑀 , that occurs after having removed the 𝑖 th row and 𝑗 th column, 𝑖 > 𝑗 + 1 , has the determinant of its ( 𝑖 βˆ’ 1 ) Γ— ( 𝑖 βˆ’ 1 ) 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 𝑖 , 𝑖 + 1 , … , 𝑛 βˆ’ 1 vanish.

References

  1. H. W. Milnes, β€œA note concerning the properties of a certain class of test matrices,” Mathematics of Computation, vol. 22, pp. 827–832, 1968. View at Zentralblatt MATH
  2. R. J. Herbold, β€œA generalization of a class of test matrices,” Mathematics of Computation, vol. 23, pp. 823–826, 1969. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. G. Carayannis, N. Kalouptsidis, and D. G. Manolakis, β€œFast recursive algorithms for a class of linear equations,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 30, no. 2, pp. 227–239, 1982.
  4. M. J. C. Gover and S. Barnett, β€œBrownian matrices: properties and extensions,” International Journal of Systems Science, vol. 17, no. 2, pp. 381–386, 1986.
  5. F. N. Valvi, β€œAnalytic inversion for a general case of certain classes of matrices,” International Journal of Computer Mathematics, vol. 72, no. 2, pp. 199–205, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet