^{1}

Nonnegativity of the Moore-Penrose inverse of a perturbation of the form

We consider the problem of characterizing nonnegativity of the Moore-Penrose inverse for matrix perturbations of the type

The main objective of the present work is to study certain structured perturbations

In this paper, first we present alternative proofs of generalizations of the SMW formula for the cases of the Moore-Penrose inverse (Theorem

Before concluding this introductory section, let us give a motivation for the work that we have undertaken here. It is a well-documented fact that

Let

Let

Let

More generally, matrices having nonnegative inverses are characterized using a property called monotonicity. The notion of monotonicity was introduced by Collatz [

One of the frequently used tools in studying monotone matrices is the notion of a regular splitting. We only refer the reader to the book [

The notion of monotonicity has been extended in a variety of ways to singular matrices using generalized inverses. First, let us briefly review the notion of two important generalized inverses.

For

The following theorem by Desoer and Whalen, which is used in the sequel, gives an equivalent definition for the Moore-Penrose inverse. Let us mention that this result was proved for operators between Hilbert spaces.

Let

Now, for

Some of the well-known properties of the Moore-Penrose inverse and the group inverse are given as follows:

In matrix analysis, a decomposition (splitting) of a matrix is considered in order to study the convergence of iterative schemes that are used in the solution of linear systems of algebraic equations. As mentioned earlier, regular splittings are useful in characterizing matrices with nonnegative inverses, whereas, proper splittings are used for studying singular systems of linear equations. Let us next recall this notion. For a matrix

Let

The following result by Berman and Plemmons [

Let

The primary objects of consideration in this paper are generalized inverses of perturbations of certain types of a matrix

Let

Set

Now,

Let

The result for the group inverse follows.

Let

Since

Suppose that

Conversely, suppose that

We conclude this section with a fairly old result [

Let

In this section, we consider perturbations of the form

Let

There exists

Let us prove the first result of this article. This extends Theorem

Let

First, we observe that since

The following result is a special case of Theorem

Let

The following consequence of Theorem

Let

From the proof of Theorem

In the rest of this section, we discuss two applications of Theorem

The first result is motivated by Theorem

For

Now, we obtain the nonnegative least element of a polyhedral set defined by a perturbed matrix. This is an immediate application of Theorem

Let

From the assumptions, using Theorem

To state and prove the result for interval matrices, let us first recall the notion of interval matrices. For

Let

In such a case, we have

Now, we present a result for the perturbation.

Let

In that case,

It follows from Theorem

In this section, we present results that typically provide conditions for iteratively defined matrices to have nonnegative Moore-Penrose inverses given that the matrices that we start with have this property. We start with the following result about the rank-one perturbation case, which is a direct consequence of Theorem

Let

Let us consider

Let

Let

Let

Set

Now, assume that

The converse part can be proved iteratively. Then condition

The following result is an extension of Lemma 2.3 in [

Let

Let us first observe that

Suppose that

Let

Now, we obtain another necessary and sufficient condition for

Let

The range conditions in (

Now, applying the above argument to the matrix

We conclude the paper by considering an extension of Example

For a fixed

Let