1. Introduction
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 [1] (Equation (2)) the explicit inverse of a class of matrices
with elements
(1)
is given. On the other hand, the analytic expressions of the inverses of two symmetric matrices
and
, where
(2)
respectively, are presented in [2] (first equation in p. 113, and Equation (1), respectively). The matrix K is a special case of Brownian matrix and
is a lower Brownian matrix, as they have been defined in [3] (Equation (2.1)). Earlier, in [4] (paragraph following Equation (3.3)) the term “pure Brownian matrix” for the type of the matrix K has introduced. Furthermore, in [5] (discussion concerning Equations (28)-(30)) the so-called “diagonal innovation matrices” (DIM) have been treated, special cases of which are the matrices K and N.
In the present paper, we consider two matrices A1 and A2 defined by
(3)
where the symbol
denotes the Hadamard product. Hence, the matrices have the forms
(4)
and
(5)
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
(6)
The matrix A1 (Equation (4)) is a lower Brownian matrix. Furthermore, the matrix PNP, where
is the permutation matrix with elements
(7)
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
.
2. The Inverse and Determinant of A1
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
(8)
where
(9)
with
(10)
and with the obvious assumptions
(11)
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):
![](https://www.scirp.org/html/16-7400911\62ab13a7-b70f-4ed1-bc1a-644885aeaea1.jpg)
which transforms A1 into the lower triangular matrix C1 given by
![](https://www.scirp.org/html/16-7400911\2e1e612f-a22a-42fb-acb3-39f7546db4e9.jpg)
and the identity matrix I into the upper bidiagonal matrix F1 with main diagonal
![](https://www.scirp.org/html/16-7400911\7356e6dd-cc27-4b99-9887-15bf57973377.jpg)
and upper first diagonal
![](https://www.scirp.org/html/16-7400911\461042bb-6b7a-4094-befc-5ecc498ced1c.jpg)
Operation 2 (applied on
and
):
![](https://www.scirp.org/html/16-7400911\cd50a412-ede4-4c6b-b2c6-a36013c5917d.jpg)
which derives a lower bidiagonal matrix
with main diagonal
![](https://www.scirp.org/html/16-7400911\3601433f-a294-4497-912d-19b5aa1251ea.jpg)
and lower first diagonal
![](https://www.scirp.org/html/16-7400911\3c4a1ae4-3e08-4533-bfb4-93ad39958e09.jpg)
while the matrix
is transformed into the tridiagonal matrix
given by
![](https://www.scirp.org/html/16-7400911\091a706d-ada4-4a21-8311-4671f7c4acf2.jpg)
Operation 3 (applied on
and
):
![](https://www.scirp.org/html/16-7400911\6a8f606e-8ab6-4612-b8ce-17c2534869f4.jpg)
which derives the diagonal matrix
![](https://www.scirp.org/html/16-7400911\9aaf0dbf-c8ae-428e-a94d-536e29e31df0.jpg)
and, respectively, the lower Hessenberg matrix F3 given by
![](https://www.scirp.org/html/16-7400911\e4f62458-5246-4e66-ae86-f463c5f07008.jpg)
with the symbol s standing for the quantity
.
Operation 4 (applied on
and
):
![](https://www.scirp.org/html/16-7400911\598cfbc3-90c9-42ed-a44c-c921b5d32610.jpg)
which transforms
into the identity matrix I and the matrix
into the inverse
.
The determinant of
takes the form
(12)
Evidently,
is singular if
or, considering the relation (9), if
for some
.
3. The Inverse and Determinant of A2
In the case of
, its inverse
is a lower Hessenberg matrix with elements given by the relations
(13)
where
(14)
with
(15)
and with the obvious assumptions
(16)
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):
![](https://www.scirp.org/html/16-7400911\79d004b5-687b-4040-9051-c55252b8607d.jpg)
which transforms A2 into the lower triangular matrix
equal to
![](https://www.scirp.org/html/16-7400911\8219177e-1bcf-44ff-90b0-f6bc7d35053f.jpg)
and the identity matrix I into the bidiagonal matrix
with main diagonal
![](https://www.scirp.org/html/16-7400911\47a0b236-c06c-4f41-9e10-97936a14f65f.jpg)
and upper first diagonal
![](https://www.scirp.org/html/16-7400911\bdec5391-997a-42cf-9423-7bd9d518db7f.jpg)
Operation 2 (applied on
and
):
![](https://www.scirp.org/html/16-7400911\f5a68aa7-b7ac-4d20-8d69-bf23d6933fe1.jpg)
which derives the lower bidiagonal matrix D2 with main diagonal
![](https://www.scirp.org/html/16-7400911\6589da8d-e119-4e57-a530-16b103877b4d.jpg)
and lower first diagonal
![](https://www.scirp.org/html/16-7400911\eaf9f765-5144-4836-a3ee-6bc40080a4a8.jpg)
while the matrix
is transformed into the tridiagonal matrix
with main diagonal
![](https://www.scirp.org/html/16-7400911\5d6f6c3c-be15-4c28-a38f-aeb03c82032d.jpg)
upper first diagonal
![](https://www.scirp.org/html/16-7400911\4ce7e7ee-b923-4ce7-8eed-bc0f331d6ca9.jpg)
and lower first diagonal
![](https://www.scirp.org/html/16-7400911\4313843a-ffed-4ff0-9829-46e4d737029a.jpg)
Operation 3 (applied on
and
):
![](https://www.scirp.org/html/16-7400911\c879d4ff-b588-4721-b9d2-e2893546e44f.jpg)
with
, which yields the diagonal matrix
,
![](https://www.scirp.org/html/16-7400911\02144315-c979-4d60-b4b4-221224fb8bae.jpg)
and the lower Hessenberg matrix
equal to
![](https://www.scirp.org/html/16-7400911\be57687c-bdcd-4b78-9f91-e2c21afb56a5.jpg)
where the symbol
stands for
.
Operation 4 (applied on
and
):
![](https://www.scirp.org/html/16-7400911\e28742ec-2a20-4674-9bb4-30fdfb43394d.jpg)
which transforms
into the identity matrix I and
into the inverse
.
The determinant of
has the form
(17)
which shows in turn that the matrix
is singular if
, or, adopting the conventions (14), if
for some
.
4. Numerical Complexity
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
(18)
(19)
(20)
(21)
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
![](https://www.scirp.org/html/16-7400911\4d70bda6-5d8b-4131-908c-c38ae644524f.jpg)
![](https://www.scirp.org/html/16-7400911\8a768180-0f01-47d3-bc1b-140a98c1fe6e.jpg)
![](https://www.scirp.org/html/16-7400911\163823b8-7a7f-46dd-8235-b0691fc87867.jpg)
![](https://www.scirp.org/html/16-7400911\7ba670f5-ca11-4c86-89c7-23a1da63fb67.jpg)
For the computation of
the algorithms (18)-(21) changes only in the estimation of the diagonal elements, for which we have
![](https://www.scirp.org/html/16-7400911\2e6f11d4-d879-4ab7-809a-8cfa71d0bf2c.jpg)
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
.
5. Concluding Remarks
The matrices A1 and A2 represent generalizations of known classes of test matrices. For instance, the test matrices given in [6] (Equations (2.1) and (2.2)) and in [1] (Eq. (2)) belong to the categories presented. Furthermore, by restricting the a’s and b’s to unity, A1 and A2 reduce to the matrices given in [2]. Also, the matrices in [7] (pp. 41, 42, 49) are special cases of A1 and A2. On the other hand, concerning the recursive algorithms given in Section 4, we have performed numerical experiments by assigning random values to the parameters of A1, and with a variety of the order n from 256 to 1024. We have found that computing
by the recursive algorithms (18)-(21) is ~100 times faster than using the LU decomposition when n = 256 and increases gradually to ~1000 times faster when n = 1024.