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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.

In many mathematical problems,

An

if

In this paper, we consider the inverse of perturbed M-matrix. Specifically we consider the effect of changing single elements inside the diagonal of

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.

In this section we introduce the notation that will be used in developing the paper. For

denoted by

Throughout this paper we use the following notation for general tridiagonal M-matrix:

where

We let

Definition 2.1 Compound Matrices ([

Let

Construct the following table which depends on

The created matrix

is called

For example, if

Then

We start with some basic facts on tridiagonal M-matrices. We can find the determinant of any

And we have the following proposition for finding the determinant of a

Proposition 3.1 ([

We will present now some of propositions of nonsingular totally nonnegative matrices which important for our work.

Proposition 3.2 [

For any nonsingular totally nonnegative matrix

That is

Proposition 3.3 ([

Let M be a nonsingular tridiagonal M-matrix, and

In the sequel we will make use the following lemma, see, e.g. [

Lemma 3.4 (Sylvester Identity)

Partition square matrix

where

Define the submatrices

If

Lemma 3.5 ([

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 [

verse matrix

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

If

tive matrix. Moreover,

Proof: Let

Then

You can find this formula in ([

There is an explicit formula for the determinant of

Multiply the first row by

where

And now apply an induction argument to get the result.

Numerical Example: Let

Note that

Numerically we can conclude the following fact.

Fact: For any

Moreover,

To prove this result we use Theorem 4.1.

Suppose M is nonsingular then

For example, when

Similarly we can find

Illustrative Example: Let

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

Proof:

Assume

Note that

Moreover we conclude the following theorem.

Theorem 4.3 Let M be the M-matrix defined above then

For example

Let

Now, we will perturb elements inside the diagonal band of the inverse of M-matrix without losing the nonnegativity property. We begin with the

Theorem 4.4 Let M be a strictly diagonally dominant tridiagonal

is totally nonnegative for all

Proof:

Let

Be a nonsingular strictly diagonally dominant tridiagonal M-matrix then

By Lemma 3.5 and Proposition 3.2, we have

By using the formula in Proposition 3.3

Note that a similar result holds for decreasing the element

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

To compute

Take the case when

Now suppose

Suppose that

which contradicts the nonnegativity of

Numerical Example: Let

The matrices

are TNN matrices.

Note that