Nonnegativity preservation under singular values perturbation.(Research Article)(Report)

1. Introduction

A singular value decomposition of a matrix A [member of] [C.sup.mxn] is a factorization A = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and both U [member of] [C.sup.mxm] and V [member of] [C.sup.nxn] are unitary. The diagonal entries of [SIGMA] are called the singular values of A. The columns [U.sub.j] of U are called left singular vectors of A and the columns [v.sub.j] of V are called right singular vectors of A. Every A e [C.sup.mxn] has a singular value decomposition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and the following relations hold: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], then U and V may be taken to be real (see [1]).

Let A be an m x n positive matrix with singular values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] 0,r = min{m,n} and left and right singular vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], respectively. In this paper we study how singular values and singular vectors of A change, under matrix perturbations of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Perturbations of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] were used in [2] to construct nonnegative matrices with prescribed extremal singular values. Both kinds of perturbations are closely related to the inverse singular value problem (ISVP), which is the problem of constructing a structured matrix from its singular values. ISVP arises in many areas of application, such as circuit theory, computed tomography, irrigation theory, mass distributions, and so forth (see [3]). The ISVP can be seen as an extension of the inverse eigenvalue problem (IEP), which look for necessary and sufficient conditions for the existence of a structured matrix with prescribed spectrum. This problem arises in different applications, see for instance [4]. When the matrix is required to be nonnegative, we have the nonnegative inverse eigenvalue problem (NIEP).

In [5, 6] and references therein, in connection with the NIEP, it was used as a perturbation result due to Brauer [7], which shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This result was extended by Rado and presented by Perfect [8] to modify r eigenvalues of a matrix of order n, r [less than or equal to] n, via a perturbation of rank-r, without changing any of the n - r remaining eigenvalues. It was also used in conection with NIEP in [8, 9]. Since the eigenvalues and singular values of a matrix are closely related, the perturbation results of this paper, which preserve nonnegativity, may be also important in the NIEP. In particular, for the symmetric case, that is, the construction of a symmetric nonnegative matrix with prescribed spectrum, since the singular values are absolute values of the eigenvalues, similar results are obtained (see [10]).In [2] the following simple singular value version of the Rado and Brauer results were given.

Theorem 1.1. Let A be an mxn matrix with singular values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or > 0,r = min{m, n}. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1.1)

be matrices of order m x p and n x p, whose columns are the left and right singular vectors, respectively, corresponding to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with [sigma].sub.i] + [d.sub.i] > 0. Then A + UD[V.sup.*] has singular values

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (1.2)

Note that the singular values of A + UD[V.sup.*] are not necessarily in nondecreasing order.

However we can reorderer them by using an appropriate permutation.

Corollary 1.2. Let A be an m x n matrix with singular values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] min{m,n}. Let ui and vi, respectively, the left and right singular vectors corresponding to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] 1,...,r.Let [alpha] [member of] R such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then A + [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] has singular values

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (1.3)

Remark 1.3. The perturbation given by Theorem 1.1 allow us to have certain control on the spectral condition number of the perturbed matrix. That is, if [[kappa].sub.2] (A)= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], then we may choose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or in such a way that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.4)

The paper is organized as follows. In Section 2 we consider perturbations of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], …