Inverse Nonnegativity of Tridiagonal M-Matrices under Diagonal Element-Wise Perturbation

One of the most important properties of M-matrices is element-wise non-negative of its inverse. In this paper, we consider element-wise perturbations of tridiagonal M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The largest interval is given by which the diagonal entries of the inverse of tridiagonal M-matrices can be perturbed without losing the property of total nonnegativity. A numerical example is given to illustrate our findings.


Introduction
In many mathematical problems, Z -matrices and M -matrices play an important role.It is often useful to know the properties of their inverses, especially when the Z -matrices and the M-matrices have a special com- binatorial 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 n n × real matrix and 0 ij m ≤ , i j ≠ , 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 0 ij a = whenever 1 i j − > ) and M is generalized strictly diagonally dominant.A matrix is said to be generalized (strictly) diagonally dominant > ∑ .Of particular importance to us is the fact that since M is an M-matrix it is non-singular and 1 M − > 0 where 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 M is an M-matrix if and only if 0 ij m ≤ , i j ≠ and 0 ii m > , and M is generalized strictly diagonally do- minant.
In this paper, we consider the inverse of perturbed M-matrix.Specifically we consider the effect of changing single elements inside the diagonal of 1 M − .We are interested in the large amount by which the single diagonal element of 1 M − 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.

Notations
In this section we introduce the notation that will be used in developing the paper.For , k n we denote by , the dispersion of α , denoted by ( ) d α , is defined to be ( ) Throughout this paper we use the following notation for general tridiagonal M-matrix: are the index sets of cardinality

Construct the following table which depends on
The created matrix , compound matrix of A .For example, if 1 2 3 4 5 6 7 8 9 .

Auxiliary Results
We start with some basic facts on tridiagonal M-matrices.We can find the determinant of any n n by using the following recursion equation [8] [9].
And we have the following proposition for finding the determinant of a n n  .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 ( ) , 1 , all principle minors are positive.That is ( ) Let M be a nonsingular tridiagonal M-matrix, and 1 M − be the inverse of the matrix M then ( ) ( ) In the sequel we will make use the following lemma, see, e.g.[12].

Lemma 3.4 (Sylvester Identity)
Partition square matrix p of order n, 2 n > , as: We now state an important result which links the determinant of M-matrix with the value of the elements of its inverse.
be a tridiagonal matrix of order n, then we can find the elements of in-  by using the following formula ( ) ( )

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 be strictly diagonally dominant M-matrix.
There is an explicit formula for the determinant of ( ) ( ) , ( ) And now apply an induction argument to get the result.

Numerical Example:
be strictly diagonally dominant M-matrix, then ( ) Fact: For any n n the following formula is true.
To prove this result we use Theorem 4.1.
Suppose M is nonsingular then ( ) ( ) For example, when 3 n = , the M-matrix of our form has an inverse given as Similarly we can find ( ) Illustrative Example: Let be a tridiagonal M-matrix and 1 0.3784 0.1351 0.0541 0.1351 0.4054 0.1622 0.0811 0.2432 0.2973 Observe that 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 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 Note that a similar result holds for decreasing the element where α α ′ ⊂ , β β ′ ⊂ , and Let M be strictly diagonally dominant M-matrix.Then ( ) Numerically we can conclude the following fact.
will perturb elements inside the diagonal band of the inverse of M-matrix without losing the nonnegativity property.We begin with the ( ) 1,1 element then generalize to other elements.Theorem 4.4 Let M be a strictly diagonally dominant tridiagonal M -matrix.Then the matrix

1 M
a nonsingular strictly diagonally dominant tridiagonal M-matrix then − is totally nonnegative.By Lemma 3.5 and Proposition 3.2, we have 1 11 this result for the other elements of diagonal.Theorem 4.5 Assume M is a strictly diagonally dominant tridiagonal M-matrix.Then the matrix 1

1 M
, detX is a positive linear compination of minors of − and hence is positive, which contradicts the assumption.

E to be the square standard basis matrix whose only nonzero entry is 1 that occurs in the ( )
)