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Advances in Numerical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 593548, 6 pages
http://dx.doi.org/10.1155/2011/593548
Research Article

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 4π‘›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(𝑛2).

1. Introduction

In [1], a class of matrices 𝐾𝑛=[π‘Žπ‘–π‘—] with elementsπ‘Žπ‘–π‘—=ξƒ―π‘Ž1,𝑖⩽𝑗,𝑗,𝑖>𝑗(1.1)is treated. A generalization of this class is presented in [2] by the matrix 𝐺𝑛=[𝑏𝑖𝑗], where 𝑏𝑖𝑗=ξƒ―π‘π‘—π‘Ž,𝑖⩽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 𝐺𝑛 and a matrix 𝐿, which results from 𝐺𝑛 first by assigning the values π‘Žπ‘–=π‘™π‘›βˆ’π‘–+1 and 𝑏𝑖=π‘˜π‘›βˆ’π‘–+1 to the latter in order to get a matrix 𝐾, say, and then by the relation 𝐿=𝑃𝐾𝑇𝑃, where 𝑃=[𝑝𝑖𝑗] is the permutation matrix with elements 𝑝𝑖𝑗=ξƒ―1,𝑖=π‘›βˆ’π‘—+1,0,otherwise.(1.3) The so-constructed class is defined by 4π‘›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 𝐺𝑛 that produce the class 𝑀=πΏβˆ˜πΊπ‘› 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‘šπ‘–π‘—] be the matrix with elements π‘šπ‘–π‘—=ξƒ―π‘˜π‘–π‘π‘—π‘™,𝑖⩽𝑗,π‘–π‘Žπ‘—,𝑖>𝑗,(2.1) that is, βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘˜π‘€=1𝑏1π‘˜1𝑏2π‘˜1𝑏3β‹―π‘˜1π‘π‘›βˆ’1π‘˜1𝑏𝑛𝑙2π‘Ž1π‘˜2𝑏2π‘˜2𝑏3β‹―π‘˜2π‘π‘›βˆ’1π‘˜2𝑏𝑛𝑙3π‘Ž1𝑙3π‘Ž2π‘˜3𝑏3β‹―π‘˜3π‘π‘›βˆ’1π‘˜3π‘π‘›β‹―π‘™π‘›βˆ’1π‘Ž1π‘™π‘›βˆ’1π‘Ž2π‘™π‘›βˆ’1π‘Ž3β‹―π‘˜π‘›βˆ’1π‘π‘›βˆ’1π‘˜π‘›βˆ’1π‘π‘›π‘™π‘›π‘Ž1π‘™π‘›π‘Ž2π‘™π‘›π‘Ž3β‹―π‘™π‘›π‘Žπ‘›βˆ’1π‘˜π‘›π‘π‘›βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(2.2) If π‘€βˆ’1=[πœ‡π‘–π‘—] is its inverse, then the following expressions give its elements πœ‡π‘–π‘—=⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©π‘˜π‘–+1π‘π‘–βˆ’1βˆ’π‘™π‘–+1π‘Žπ‘–βˆ’1π‘π‘–βˆ’1π‘π‘–π‘˜,𝑖=𝑗=2,3,…,π‘›βˆ’1,2𝑐0𝑐1𝑏,𝑖=𝑗=1,π‘›βˆ’1π‘π‘›βˆ’1𝑐𝑛(,𝑖=𝑗=𝑛,βˆ’1)𝑖+π‘—π‘‘π‘—βˆ’1π‘”π‘–βˆπ‘–βˆ’1𝜈=𝑗+1π‘“πœˆβˆπ‘–πœˆ=π‘—βˆ’1π‘πœˆβˆ’1,𝑖>𝑗,𝑐𝑖,𝑖=π‘—βˆ’1,0,𝑖<π‘—βˆ’1,(2.3) where 𝑐𝑖=π‘˜π‘–+1π‘π‘–βˆ’π‘™π‘–+1π‘Žπ‘–,𝑖=1,2,…,π‘›βˆ’1,𝑐0=π‘˜1,𝑐𝑛=𝑏𝑛,𝑑𝑖=π‘Žπ‘–+1π‘π‘–βˆ’π‘Žπ‘–π‘π‘–+1,𝑖=1,2,…,π‘›βˆ’2,𝑑0=π‘Ž1,𝑓𝑖=π‘™π‘–π‘Žπ‘–βˆ’π‘˜π‘–π‘π‘–π‘”,𝑖=2,3,…,π‘›βˆ’1,𝑖=π‘˜π‘–+1π‘™π‘–βˆ’π‘˜π‘–π‘™π‘–+1,𝑖=2,3,…,π‘›βˆ’1,𝑔𝑛=𝑙𝑛,(2.4) with π‘–βˆ’1ξ‘πœˆ=𝑗+1π‘“πœˆ=1whenever𝑖=𝑗+1,(2.5) and with the obvious assumption 𝑐𝑖≠0,𝑖=0,1,2,…,𝑛.(2.6)

3. The Proof

We prove that the expressions (2.3) give the inverse matrix π‘€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. rowπ‘–βˆ’(π‘˜π‘–/π‘˜π‘–+1)Γ—row(𝑖+1),𝑖=1,2,…,π‘›βˆ’1, which gives the lower triangular matrix βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘˜1π‘˜2𝑐1π‘Ž0β‹―001π‘˜3𝑔2π‘˜2π‘˜3𝑐2π‘Žβ‹―00β‹―β‹―β‹―β‹―β‹―1π‘˜π‘›π‘”π‘›βˆ’1π‘Ž2π‘˜π‘›π‘”π‘›βˆ’1β‹―π‘˜π‘›βˆ’1π‘˜π‘›π‘π‘›βˆ’10π‘”π‘›π‘Ž1π‘”π‘›π‘Ž2β‹―π‘”π‘›π‘Žπ‘›βˆ’1π‘”π‘›π‘π‘›βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.1)

Operation. rowπ‘–βˆ’(π‘˜π‘–π‘”π‘–/π‘˜π‘–+1π‘”π‘–βˆ’1)Γ—row(π‘–βˆ’1),𝑖=𝑛,π‘›βˆ’1,…,3,π‘˜π‘›+1=1, which results in a bidiagonal matrix with main diagonal ξ‚΅π‘˜1𝑐1π‘˜2,π‘˜2𝑐2π‘˜3π‘˜,…,π‘›βˆ’1π‘π‘›βˆ’1π‘˜π‘›,π‘˜π‘›π‘π‘›ξ‚Ά(3.2) and lower first diagonal ξ‚΅π‘Ž1𝑔2π‘˜3,π‘˜3𝑔3𝑓2π‘˜4𝑔2π‘˜,…,π‘›βˆ’1π‘”π‘›βˆ’1π‘“π‘›βˆ’2π‘˜π‘›π‘”π‘›βˆ’2,π‘˜π‘›π‘”π‘›π‘“π‘›βˆ’1gπ‘›βˆ’1ξ‚Ά.(3.3)

Operation. row2βˆ’(π‘˜2π‘Ž1𝑔2/π‘˜1π‘˜3𝑐1)Γ—row1,androwπ‘–βˆ’(π‘˜π‘–π‘˜π‘–π‘”π‘–π‘“π‘–βˆ’1/π‘˜π‘–βˆ’1π‘˜π‘–+1π‘”π‘–βˆ’1π‘π‘–βˆ’1)Γ—row(π‘–βˆ’1),𝑖=3,4,…,𝑛,which gives the diagonal matrix ξƒ‘π‘˜1𝑐1π‘˜2π‘˜2𝑐2π‘˜3β‹―π‘˜π‘›βˆ’1π‘π‘›βˆ’1π‘˜π‘›π‘˜π‘›π‘π‘›ξƒ”.(3.4)

Operation. π‘˜π‘–+1/π‘˜π‘–π‘π‘–Γ—row𝑖,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‘˜1π‘˜2π‘˜,βˆ’2π‘˜3π‘˜,…,βˆ’π‘›βˆ’1π‘˜π‘›ξ‚Ά.(3.6)

Form. The tridiagonal matrix βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘˜1βˆ’1π‘˜2π‘˜0β‹―0001βˆ’2π‘˜3π‘˜β‹―000βˆ’3𝑔3π‘˜4𝑔2π‘˜3ξ€·π‘˜4𝑙2βˆ’π‘˜2𝑙4ξ€Έπ‘˜4𝑔2β‹―π‘˜β‹―00000β‹―π‘›βˆ’1ξ€·π‘˜π‘›π‘™π‘›βˆ’2βˆ’π‘˜π‘›βˆ’2π‘™π‘›ξ€Έπ‘˜π‘›π‘”π‘›βˆ’2βˆ’π‘˜π‘›βˆ’1π‘˜π‘›π‘˜000β‹―βˆ’π‘›π‘”π‘›π‘”π‘›βˆ’1π‘˜π‘›π‘™π‘›βˆ’1π‘”π‘›βˆ’1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.7)

Form. The lower Hessenberg matrix βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘˜1βˆ’1π‘˜2βˆ’π‘Žβ‹―001𝑔2π‘˜2π‘˜1π‘˜3𝑐1π‘˜2π‘˜3𝑐1ξ€·π‘˜3𝑏1βˆ’π‘™3π‘Ž1ξ€Έπ‘Žβ‹―001𝑔3π‘˜3𝑓2π‘˜1π‘˜4𝑐1𝑐2βˆ’π‘‘1𝑔3π‘˜3π‘˜4𝑐1𝑐2β‹―π‘ π‘Žβ‹―001π‘”π‘›βˆ’1π‘˜π‘›βˆ’1𝑓2β‹―π‘“π‘›βˆ’2π‘˜1π‘˜π‘›π‘1β‹―π‘π‘›βˆ’2𝑠𝑑1π‘”π‘›βˆ’1π‘˜π‘›βˆ’1𝑓3β‹―π‘“π‘›βˆ’2π‘˜π‘›π‘1𝑐2β‹―π‘π‘›βˆ’2β‹―π‘˜π‘›βˆ’1kπ‘›π‘π‘›βˆ’2ξ€·π‘˜π‘›π‘π‘›βˆ’2βˆ’π‘™π‘›π‘Žπ‘›βˆ’2ξ€Έβˆ’π‘˜π‘›βˆ’1π‘˜π‘›π‘ π‘Ž1π‘”π‘›π‘˜π‘›π‘“2β‹―π‘“π‘›βˆ’1π‘˜1𝑐1β‹―π‘π‘›βˆ’1𝑠𝑑1π‘”π‘›π‘˜π‘›π‘“3β‹―π‘“π‘›βˆ’1𝑐1𝑐2β‹―π‘π‘›βˆ’1π‘‘β‹―βˆ’π‘›βˆ’2π‘”π‘›π‘˜π‘›π‘π‘›βˆ’2π‘π‘›βˆ’1π‘π‘›βˆ’1π‘˜π‘›π‘π‘›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 det(𝑀)=π‘˜1π‘π‘›ξ€·π‘˜2𝑏1βˆ’π‘™2π‘Ž1ξ€Έβ‹―ξ€·π‘˜π‘›π‘π‘›βˆ’1βˆ’π‘™π‘›π‘Žπ‘›βˆ’1ξ€Έ.(3.9) Evidently, 𝑀 is singular if 𝑐𝑖=0 for some π‘–βˆˆ{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 π‘€βˆ’1. In particular, πœ‡π‘–,π‘–βˆ’1𝑑=βˆ’π‘–βˆ’2π‘”π‘–π‘π‘–βˆ’2π‘π‘–βˆ’1π‘π‘–πœ‡,𝑖=2,3,…,𝑛,𝑖,π‘–βˆ’π‘ βˆ’1𝑑=βˆ’π‘–βˆ’π‘ βˆ’2π‘“π‘–βˆ’π‘ π‘‘π‘–βˆ’π‘ βˆ’1π‘π‘–βˆ’π‘ βˆ’2πœ‡π‘–,π‘–βˆ’π‘ ,𝑖=3,4,…,𝑛,𝑠=1,2,…,π‘–βˆ’2,(4.1)

or, alternatively, πœ‡π‘—+1,𝑗𝑑=βˆ’π‘—βˆ’1𝑔𝑗+1π‘π‘—βˆ’1𝑐𝑗𝑐𝑗+1πœ‡,𝑗=1,2,…,π‘›βˆ’1,𝑗+𝑠+1,𝑗𝑔=βˆ’π‘—+𝑠+1𝑓𝑗+𝑠𝑔𝑗+𝑠𝑐𝑗+𝑠+1πœ‡π‘—+𝑠,𝑗,𝑗=1,2,…,π‘›βˆ’2,𝑠=1,2,…,π‘›βˆ’π‘—βˆ’1,(4.2)

where the 𝑐𝑖, 𝑑𝑖, 𝑓𝑖, and 𝑔𝑖 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 2𝑛2+11π‘›βˆ’19 multiplications/divisions, since the coefficient of πœ‡π‘–π‘— depends only on the second (first) subscript, respectively, and in 5π‘›βˆ’9 additions/subtractions.

5. Remarks

When replacing π‘˜π‘–, 𝑏𝑖, and π‘Žπ‘– by π‘Žπ‘–, π‘˜π‘–, and 𝑏𝑖, respectively, the matrix 𝑀 (see (2.2)) is transformed into the transpose 𝐢𝑇 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‘šπ‘–π‘—,𝑖>𝑗+1, vanish to provide the lower Hessenberg type inverse. In detail, each minor of 𝑀, that occurs after having removed the 𝑖th row and 𝑗th column, 𝑖>𝑗+1, has the determinant of its (π‘–βˆ’1)Γ—(π‘–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,…,π‘›βˆ’1 vanish.

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