1. Introduction
In many mathematical problems,
-matrices and
-matrices play an important role. It is often useful to know the properties of their inverses, especially when the
-matrices and the M-matrices have a special combinatorial structure, for more details we refer the reader [1] . M-matrices have important applications, for instance, in iterative methods, in numerical analysis, in the analysis of dynamical systems, in economics, and in mathematical programming. One of the most important properties of some kinds of M-matrices is the nonegativity of their inverses, which plays central role in many of mathematical problems.
An
real matrix
is called M-matrix if
,
and
,
, over the years, M-matrices have considerable attention, in large part because they arise in many applications [2] [3] . Recently, a noticeable amount of attention has turned to the inverse of tridiagonal M-matrices (those matrices which happen to be inverses of special form of M-matrices with property
whenever
) and
is generalized strictly diagonally dominant. A matrix is said to be generalized (strictly) diagonally dominant
if
. Of particular importance to us is the fact that since
is an M-matrix it is non-singular and
>0where the inequality is satisfied element-wise. A rich class of M-matrices were introduced by Ostrowski in 1937 [4] , with reference to the work of Minkowski [5] [6] . A condition which is easy to check is that a matrix
is an M-matrix if and only if
,
and
, and
is generalized strictly diagonally dominant.
In this paper, we consider the inverse of perturbed M-matrix. Specifically we consider the effect of changing single elements inside the diagonal of
. We are interested in the large amount by which the single diagonal element of
can be varied without losing the property of total nonnegativity.
The reminder of the paper is organized as follows. In section 2, we explain our notations and some needed important definitions are presented. In section 3, some auxiliary results and important prepositions and lemmas are stated. In section 4, we present our results.
2. Notations
In this section we introduce the notation that will be used in developing the paper. For
we denote by
the set of all strictly increasing sequences of
integers chosen from
. For
,
, we denote by
the
submatrix of
contained in the rows indexed by
and columns indexed by
. A matrix
is called totally positive (abbreviated TP henceforth) if
and totally nonnegative (abbreviated TN) if
for all
,
. For a given index
, with property
,
, the dispersion of
,
denoted by
, is defined to be
.
Throughout this paper we use the following notation for general tridiagonal M-matrix:
![]()
where
,
and
, and each
is large enough that
is strictly diagonally dominant.
We let
to be the square standard basis matrix whose only nonzero entry is 1 that occurs in the
position.
Definition 2.1 Compound Matrices ([7] , p. 19).
Let
be a square matrix of order
. Let
be the index set of cardinality
, defining
,
are the index sets of cardinality
.
Construct the following table which depends on
.
The created matrix ![]()
is called
, compound matrix of
.
For example, if
with indexed sets
,
and
.
Then
.
3. Auxiliary Results
We start with some basic facts on tridiagonal M-matrices. We can find the determinant of any
tridiagonal M-matrix
by using the following recursion equation [8] [9] .
![]()
And we have the following proposition for finding the determinant of a
tridiagonal M-matrix
.
Proposition 3.1 ([10] , formula 4.1) For any
tridiagonal M-matrix
the following relation is true
.
We will present now some of propositions of nonsingular totally nonnegative matrices which important for our work.
Proposition 3.2 [10] [11]
For any nonsingular totally nonnegative matrix
, all principle minors are positive.
That is
, for all
and
.
Proposition 3.3 ([7] , p. 21)
Let M be a nonsingular tridiagonal M-matrix, and
be the inverse of the matrix M then
, when.
In the sequel we will make use the following lemma, see, e.g. [12] .
Lemma 3.4 (Sylvester Identity)
Partition square matrix
of order n,
, as:
,
where
square matrix of order
and
,
,
and
are scalars.
Define the submatrices
![]()
If
is nonsingular, then
![]()
Lemma 3.5 ([11] , p.199) Let
be a square matrix of order
, with
. Then
is totally nonnegative
.
We now state an important result which links the determinant of M-matrix with the value of the elements of its inverse.
Lemma 3.6 [10] Let
be a tridiagonal matrix of order n, then we can find the elements of in-
verse matrix
by using the following formula
.
4. Main Results
In this section, we present our results based on the inverse of tridiagonal M-matrices. Firstly we begin with the following theorem.
Theorem 4.1
Let
be strictly diagonally dominant M-matrix.
If
is the
compound matrix of M then the matrix
is totally nonnega-
tive matrix. Moreover,
where ![]()
Proof: Let
be strictly diagonally dominant M-matrix.
Then
is totally nonnegative matrix. So is
.
You can find this formula in ([7] , p. 21).
There is an explicit formula for the determinant of
given as
![]()
Multiply the first row by
and add it to the
row to obtain
![]()
where
,
And now apply an induction argument to get the result.
Numerical Example: Let
be strictly diagonally dominant M-matrix, then
and
is totally nonnegative.
Note that
and ![]()
![]()
Numerically we can conclude the following fact.
Fact: For any
tridiagonal M-matrix
the following formula is true.
for
Moreover, ![]()
To prove this result we use Theorem 4.1.
Suppose M is nonsingular then
, so
![]()
For example, when
, the M-matrix of our form
has an inverse given as
![]()
![]()
Similarly we can find
.
Illustrative Example: Let
be a tridiagonal M-matrix and
![]()
Note that ![]()
Observe that the error came from the rounded to the nearest part of 10,000.
Theorem 4.2 Let M be a strictly diagonally dominant M-matrix, if
,
,
then
![]()
Proof:
Assume
and
, , ,.
Note that
by previous fact, and by using Sylvester's identity, we have
![]()
Moreover we conclude the following theorem.
Theorem 4.3 Let M be the M-matrix defined above then
![]()
For example
,
Let
then ![]()
and
Now, we will perturb elements inside the diagonal band of the inverse of M-matrix without losing the nonnegativity property. We begin with the
element then generalize to other elements.
Theorem 4.4 Let M be a strictly diagonally dominant tridiagonal
-matrix. Then the matrix ![]()
is totally nonnegative for all
.
Proof:
Let ![]()
Be a nonsingular strictly diagonally dominant tridiagonal M-matrix then
is totally nonnegative.
By Lemma 3.5 and Proposition 3.2, we have
is totally nonnegative
.
By using the formula in Proposition 3.3
.
Note that a similar result holds for decreasing the element
by considering the matrix
, which reverses the matrix
as the relation
.
We can generalize this result for the other elements of diagonal.
Theorem 4.5 Assume M is a strictly diagonally dominant tridiagonal M-matrix. Then the matrix ![]()
is totally nonnegative for all
.
Proof: Suppose that
is not totally nonnegative for all
, then there exist
both contain
such that
.
To compute
expand the determinant along the
row of
then
where
,
, and
is some minor of
.
Take the case when
odd. Thus,
is a positive linear compination of minors of
and hence is positive, which contradicts the assumption.
Now suppose
such that
is totally nonnegative matrix.
Suppose that
, then by Theorem 4.3.
, since
which contradicts the nonnegativity of
.
Numerical Example: Let
is strictly diagonally dominant tridiagonal M-matrix
so
The matrices
, ,
, ,
, , and
,.
are TNN matrices.
Note that
.