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 4𝑛−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 𝒪(𝑛2).

1. Introduction

In [1], a class of matrices 𝐾𝑛=[𝑎𝑖𝑗] with elements𝑎𝑖𝑗=𝑎1,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=𝑏𝑗𝑎,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 𝑎𝑖=𝑙𝑛−𝑖+1 and 𝑏𝑖=𝑘𝑛−𝑖+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=1,𝑖=𝑛−𝑗+1,0,otherwise.(1.3) The so-constructed class is defined by 4𝑛−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=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].

2. The Class of Matrices and Its Inverse

Let 𝑀=[𝑚𝑖𝑗] be the matrix with elements 𝑚𝑖𝑗=𝑘𝑖𝑏𝑗𝑙,𝑖⩽𝑗,𝑖𝑎𝑗,𝑖>𝑗,(2.1) that is, ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑘𝑀=1𝑏1𝑘1𝑏2𝑘1𝑏3⋯𝑘1𝑏𝑛−1𝑘1𝑏𝑛𝑙2𝑎1𝑘2𝑏2𝑘2𝑏3⋯𝑘2𝑏𝑛−1𝑘2𝑏𝑛𝑙3𝑎1𝑙3𝑎2𝑘3𝑏3⋯𝑘3𝑏𝑛−1𝑘3𝑏𝑛⋯𝑙𝑛−1𝑎1𝑙𝑛−1𝑎2𝑙𝑛−1𝑎3⋯𝑘𝑛−1𝑏𝑛−1𝑘𝑛−1𝑏𝑛𝑙𝑛𝑎1𝑙𝑛𝑎2𝑙𝑛𝑎3⋯𝑙𝑛𝑎𝑛−1𝑘𝑛𝑏𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(2.2) If 𝑀−1=[𝜇𝑖𝑗] is its inverse, then the following expressions give its elements 𝜇𝑖𝑗=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩𝑘𝑖+1𝑏𝑖−1−𝑙𝑖+1𝑎𝑖−1𝑐𝑖−1𝑐𝑖𝑘,𝑖=𝑗=2,3,…,𝑛−1,2𝑐0𝑐1𝑏,𝑖=𝑗=1,𝑛−1𝑐𝑛−1𝑐𝑛(,𝑖=𝑗=𝑛,−1)𝑖+𝑗𝑑𝑗−1𝑔𝑖∏𝑖−1𝜈=𝑗+1𝑓𝜈∏𝑖𝜈=𝑗−1𝑐𝜈−1,𝑖>𝑗,𝑐𝑖,𝑖=𝑗−1,0,𝑖<𝑗−1,(2.3) where 𝑐𝑖=𝑘𝑖+1𝑏𝑖−𝑙𝑖+1𝑎𝑖,𝑖=1,2,…,𝑛−1,𝑐0=𝑘1,𝑐𝑛=𝑏𝑛,𝑑𝑖=𝑎𝑖+1𝑏𝑖−𝑎𝑖𝑏𝑖+1,𝑖=1,2,…,𝑛−2,𝑑0=𝑎1,𝑓𝑖=𝑙𝑖𝑎𝑖−𝑘𝑖𝑏𝑖𝑔,𝑖=2,3,…,𝑛−1,𝑖=𝑘𝑖+1𝑙𝑖−𝑘𝑖𝑙𝑖+1,𝑖=2,3,…,𝑛−1,𝑔𝑛=𝑙𝑛,(2.4) with 𝑖−1𝜈=𝑗+1𝑓𝜈=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 𝑀−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. row𝑖−(𝑘𝑖/𝑘𝑖+1)×row(𝑖+1),𝑖=1,2,…,𝑛−1, which gives the lower triangular matrix ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑘1𝑘2𝑐1𝑎0⋯001𝑘3𝑔2𝑘2𝑘3𝑐2𝑎⋯00⋯⋯⋯⋯⋯1𝑘𝑛𝑔𝑛−1𝑎2𝑘𝑛𝑔𝑛−1⋯𝑘𝑛−1𝑘𝑛𝑐𝑛−10𝑔𝑛𝑎1𝑔𝑛𝑎2⋯𝑔𝑛𝑎𝑛−1𝑔𝑛𝑏𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(3.1)

Operation. row𝑖−(𝑘𝑖𝑔𝑖/𝑘𝑖+1𝑔𝑖−1)×row(𝑖−1),𝑖=𝑛,𝑛−1,…,3,𝑘𝑛+1=1, which results in a bidiagonal matrix with main diagonal 𝑘1𝑐1𝑘2,𝑘2𝑐2𝑘3𝑘,…,𝑛−1𝑐𝑛−1𝑘𝑛,𝑘𝑛𝑏𝑛(3.2) and lower first diagonal 𝑎1𝑔2𝑘3,𝑘3𝑔3𝑓2𝑘4𝑔2𝑘,…,𝑛−1𝑔𝑛−1𝑓𝑛−2𝑘𝑛𝑔𝑛−2,𝑘𝑛𝑔𝑛𝑓𝑛−1g𝑛−1.(3.3)

Operation. row2−(𝑘2𝑎1𝑔2/𝑘1𝑘3𝑐1)×row1,androw𝑖−(𝑘𝑖𝑘𝑖𝑔𝑖𝑓𝑖−1/𝑘𝑖−1𝑘𝑖+1𝑔𝑖−1𝑐𝑖−1)×row(𝑖−1),𝑖=3,4,…,𝑛,which gives the diagonal matrix 𝑘1𝑐1𝑘2𝑘2𝑐2𝑘3⋯𝑘𝑛−1𝑐𝑛−1𝑘𝑛𝑘𝑛𝑏𝑛.(3.4)

Operation. 𝑘𝑖+1/𝑘𝑖𝑐𝑖×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 −𝑘1𝑘2𝑘,−2𝑘3𝑘,…,−𝑛−1𝑘𝑛.(3.6)

Form. The tridiagonal matrix ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑘1−1𝑘2𝑘0⋯0001−2𝑘3𝑘⋯000−3𝑔3𝑘4𝑔2𝑘3𝑘4𝑙2−𝑘2𝑙4𝑘4𝑔2⋯𝑘⋯00000⋯𝑛−1𝑘𝑛𝑙𝑛−2−𝑘𝑛−2𝑙𝑛𝑘𝑛𝑔𝑛−2−𝑘𝑛−1𝑘𝑛𝑘000⋯−𝑛𝑔𝑛𝑔𝑛−1𝑘𝑛𝑙𝑛−1𝑔𝑛−1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(3.7)

Form. The lower Hessenberg matrix ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑘1−1𝑘2−𝑎⋯001𝑔2𝑘2𝑘1𝑘3𝑐1𝑘2𝑘3𝑐1𝑘3𝑏1−𝑙3𝑎1𝑎⋯001𝑔3𝑘3𝑓2𝑘1𝑘4𝑐1𝑐2−𝑑1𝑔3𝑘3𝑘4𝑐1𝑐2⋯𝑠𝑎⋯001𝑔𝑛−1𝑘𝑛−1𝑓2⋯𝑓𝑛−2𝑘1𝑘𝑛𝑐1⋯𝑐𝑛−2𝑠𝑑1𝑔𝑛−1𝑘𝑛−1𝑓3⋯𝑓𝑛−2𝑘𝑛𝑐1𝑐2⋯𝑐𝑛−2⋯𝑘𝑛−1k𝑛𝑐𝑛−2𝑘𝑛𝑏𝑛−2−𝑙𝑛𝑎𝑛−2−𝑘𝑛−1𝑘𝑛𝑠𝑎1𝑔𝑛𝑘𝑛𝑓2⋯𝑓𝑛−1𝑘1𝑐1⋯𝑐𝑛−1𝑠𝑑1𝑔𝑛𝑘𝑛𝑓3⋯𝑓𝑛−1𝑐1𝑐2⋯𝑐𝑛−1𝑑⋯−𝑛−2𝑔𝑛𝑘𝑛𝑐𝑛−2𝑐𝑛−1𝑏𝑛−1𝑘𝑛𝑐𝑛−1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.8) where the symbol 𝑠 stands for the quantity (−1)𝑖+𝑗.

Form. The matrix whose elements are given by the expressions (2.4).
The determinant of 𝑀 takes the form det(𝑀)=𝑘1𝑏𝑛𝑘2𝑏1−𝑙2𝑎1⋯𝑘𝑛𝑏𝑛−1−𝑙𝑛𝑎𝑛−1.(3.9) Evidently, 𝑀 is singular if 𝑐𝑖=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−1. In particular, 𝜇𝑖,𝑖−1𝑑=−𝑖−2𝑔𝑖𝑐𝑖−2𝑐𝑖−1𝑐𝑖𝜇,𝑖=2,3,…,𝑛,𝑖,𝑖−𝑠−1𝑑=−𝑖−𝑠−2𝑓𝑖−𝑠𝑑𝑖−𝑠−1𝑐𝑖−𝑠−2𝜇𝑖,𝑖−𝑠,𝑖=3,4,…,𝑛,𝑠=1,2,…,𝑖−2,(4.1)

or, alternatively, 𝜇𝑗+1,𝑗𝑑=−𝑗−1𝑔𝑗+1𝑐𝑗−1𝑐𝑗𝑐𝑗+1𝜇,𝑗=1,2,…,𝑛−1,𝑗+𝑠+1,𝑗𝑔=−𝑗+𝑠+1𝑓𝑗+𝑠𝑔𝑗+𝑠𝑐𝑗+𝑠+1𝜇𝑗+𝑠,𝑗,𝑗=1,2,…,𝑛−2,𝑠=1,2,…,𝑛−𝑗−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+11𝑛−19 multiplications/divisions, since the coefficient of 𝜇𝑖𝑗 depends only on the second (first) subscript, respectively, and in 5𝑛−9 additions/subtractions.

5. Remarks

When replacing 𝑘𝑖, 𝑏𝑖, and 𝑎𝑖 by 𝑎𝑖, 𝑘𝑖, and 𝑏𝑖, respectively, the matrix 𝑀 (see (2.2)) is transformed into the transpose 𝐶𝑇 of the matrix 𝐶 [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 𝑚𝑖𝑗,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 (𝑖−1)×(𝑖−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−1 vanish.