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

Authors: F. N. Valvi
Journal: Advances in Numerical Analysis
 (2011)

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## 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 by4n−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).


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## Body

## 1. Introduction

In [1], a class of matrices Kn=[aij] with elements(1.1)aij={1,i⩽j,aj,i>jis treated. A generalization of this class is presented in [2] by the matrix Gn=[bij], where
(1.2)bij={bj,i⩽j,aj,i>j.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
(1.3)pij={1,i=n-j+1,0,otherwise.
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 classesL 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

LetM=[mij] be the matrix with elements
(2.1)mij={kibj,i⩽j,liaj,i>j,
that is,
(2.2)M=[k1b1k1b2k1b3⋯k1bn-1k1bnl2a1k2b2k2b3⋯k2bn-1k2bnl3a1l3a2k3b3⋯k3bn-1k3bn⋯ln-1a1ln-1a2ln-1a3⋯kn-1bn-1kn-1bnlna1lna2lna3⋯lnan-1knbn].
If M-1=[μij] is its inverse, then the following expressions give its elements
(2.3)μij={ki+1bi-1-li+1ai-1ci-1ci,i=j=2,3,…,n-1,k2c0c1,i=j=1,bn-1cn-1cn,i=j=n,(-1)i+jdj-1gi∏ν=j+1i-1fν∏ν=j-1icν,i>j,-1ci,i=j-1,0,i<j-1,
where
(2.4)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-kibi,i=2,3,…,n-1,gi=ki+1li-kili+1,i=2,3,…,n-1,gn=ln,
with
(2.5)∏ν=j+1i-1fν=1wheneveri=j+1,
and with the obvious assumption
(2.6)ci≠0,i=0,1,2,…,n.


## 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
(3.1)[k1k2c10⋯00a1k3g2k2k3c2⋯00⋯⋯⋯⋯⋯a1kngn-1a2kngn-1⋯kn-1kncn-10gna1gna2⋯gnan-1gnbn].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
(3.2)(k1c1k2,k2c2k3,…,kn-1cn-1kn,knbn)
and lower first diagonal
(3.3)(a1g2k3,k3g3f2k4g2,…,kn-1gn-1fn-2kngn-2,kngnfn-1gn-1).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
(3.4)⌈k1c1k2k2c2k3⋯kn-1cn-1knknbn⌋.Operation.
ki+1/kici×rowi,i=1,2,…,n,  which gives the identity matrix.Operations1–4 transform the identity matrix to the following forms, respectively:Form.
The upper bidiagonal matrix consisting of the main diagonal(3.5)(1,1,…,1)
and the upper first diagonal
(3.6)(-k1k2,-k2k3,…,-kn-1kn).Form.
The tridiagonal matrix(3.7)[1-k1k20⋯0001-k2k3⋯000-k3g3k4g2k3(k4l2-k2l4)k4g2⋯00⋯000⋯kn-1(knln-2-kn-2ln)kngn-2-kn-1kn000⋯-kngngn-1knln-1gn-1].Form.
The lower Hessenberg matrix(3.8)[1-k1k2⋯00-a1g2k2k1k3c1k2k3c1(k3b1-l3a1)⋯00a1g3k3f2k1k4c1c2-d1g3k3k4c1c2⋯00⋯sa1gn-1kn-1f2⋯fn-2k1knc1⋯cn-2sd1gn-1kn-1f3⋯fn-2knc1c2⋯cn-2⋯kn-1kncn-2(knbn-2-lnan-2)-kn-1knsa1gnknf2⋯fn-1k1c1⋯cn-1sd1gnknf3⋯fn-1c1c2⋯cn-1⋯-dn-2gnkncn-2cn-1bn-1kncn-1],
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 ofM takes the form
(3.9)det⁡(M)=k1bn(k2b1-l2a1)⋯(knbn-1-lnan-1).
Evidently, M is singular if ci=0 for some i∈{0,1,2,…,n}.


## 4. Numerical Complexity

The inverse of the matrixM 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,
(4.1)μi,i-1=-di-2gici-2ci-1ci,i=2,3,…,n,μi,i-s-1=-di-s-2fi-sdi-s-1ci-s-2μi,i-s,i=3,4,…,n,s=1,2,…,i-2,or, alternatively,(4.2)μj+1,j=-dj-1gj+1cj-1cjcj+1,j=1,2,…,n-1,μj+s+1,j=-gj+s+1fj+sgj+scj+s+1μj+s,j,j=1,2,…,n-2,s=1,2,…,n-j-1,where theci, 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 replacingki, 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.



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*Source: 593548-2011-12-01.xml*