We present explicit inverses of two Brownian-type matrices, which are defined as Hadamard products of certain already known matrices. The matrices under consideration are defined by 3n - 1 parameters and their lower Hessenberg form inverses are expressed analytically in terms of these parameters. Such matrices are useful in the theory of digital signal processing and in testing matrix inversion algorithms.
Brownian matrices are frequently involved in problems concerning “digital signal processing”. In particular, Brownian motion is one of the most common linear models used for representing nonstationary signals. The covariance matrix of a discrete-time Brownian motion has, in turn, a very characteristic structure, the so-called “Brownian matrix”.
In [
is given. On the other hand, the analytic expressions of the inverses of two symmetric matrices and, where
respectively, are presented in [
In the present paper, we consider two matrices A1 and A2 defined by
where the symbol denotes the Hadamard product. Hence, the matrices have the forms
and
Let us now define for a matrix the terms “pure upper Brownian matrix” and “pure lower Brownian matrix”, for the elements of which the following relations are respectively valid
The matrix A1 (Equation (4)) is a lower Brownian matrix. Furthermore, the matrix PNP, where is the permutation matrix with elements
is a pure Brownian matrix and a pure lower Brownian matrix. Hence, their Hadamard product gives a pure lower Brownian matrix, that is, the matrix.
In the following sections, we deduce in analytic form the inverses and determinants of the matrices A1 and A2; and we study the numerical complexity on evaluating and.
The inverse of A1 is a lower Hessenberg matrix expressed analytically by the 3n − 1 parameters defining A1. In particular, the inverse has elements given by the relations
where
with
and with the obvious assumptions
To prove that the relations (8)-(10) give the inverse matrix, we reduce A1 to the identity matrix I by applying a number of elementary row transformations.
Then the product of the corresponding elementary matrices gives the inverse matrix of A1. These transformations are defined by the following sequence of row operations.
Operation 1 (applied on A1 and on the identity matrix I):
which transforms A1 into the lower triangular matrix C1 given by
and the identity matrix I into the upper bidiagonal matrix F1 with main diagonal
and upper first diagonal
Operation 2 (applied on and):
which derives a lower bidiagonal matrix with main diagonal
and lower first diagonal
while the matrix is transformed into the tridiagonal matrix given by
Operation 3 (applied on and):
which derives the diagonal matrix
and, respectively, the lower Hessenberg matrix F3 given by
with the symbol s standing for the quantity.
Operation 4 (applied on and):
which transforms into the identity matrix I and the matrix into the inverse.
The determinant of takes the form
Evidently, is singular if or, considering the relation (9), if for some.
In the case of, its inverse is a lower Hessenberg matrix with elements given by the relations
where
with
and with the obvious assumptions
In order to prove that the relations (13)-(15) give the inverse matrix, we follow a similar manner to that of Section 2.
Operation 1 (applied on A2 and on the identity matrix I):
which transforms A2 into the lower triangular matrix equal to
and the identity matrix I into the bidiagonal matrix with main diagonal
and upper first diagonal
Operation 2 (applied on and):
which derives the lower bidiagonal matrix D2 with main diagonal
and lower first diagonal
while the matrix is transformed into the tridiagonal matrix with main diagonal
upper first diagonal
and lower first diagonal
Operation 3 (applied on and):
with, which yields the diagonal matrix,
and the lower Hessenberg matrix equal to
where the symbol stands for.
Operation 4 (applied on and):
which transforms into the identity matrix I and into the inverse.
The determinant of has the form
which shows in turn that the matrix is singular if, or, adopting the conventions (14), if for some.
The relations (8) and (13) lead to recurrence formulae, by which the inverses and, respectively, are computed in multiplications/divisions and additions/substractions. In fact, the recursive algorithm
where, , , and are given by the relation (9), computes in mult/div (since the coefficients of depends only on the second subscript) and add/sub.
In terms of, the above algorithm takes the form
For the computation of the algorithms (18)-(21) changes only in the estimation of the diagonal elements, for which we have
where, , , and are given by the relation (14). Therefore, considering the relations (9) and (14), it is clear that the number of mult/div and add/sub in computing is the same with that of.
The matrices A1 and A2 represent generalizations of known classes of test matrices. For instance, the test matrices given in [