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Mathematical Problems in Engineering
Volumeย 2009, Article IDย 301582, 25 pages
http://dx.doi.org/10.1155/2009/301582
Research Article

Nonnegativity Preservation under Singular Values Perturbation

1Departamento de Matemรกticas, Universidad Catรณlica del Norte, Casilla 1280. Antofagasta, Chile
2Departamento de Matemรกtica y Fรญsica, Universidad de Magallanes, Casilla 113-D, Punta Arenas, Chile

Received 30 December 2008; Accepted 18 May 2009

Academic Editor: Davidย Chelidze

Copyright ยฉ 2009 Emedin Montaรฑo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study how singular values and singular vectors of a matrix ๐ด change, under matrix perturbations of the form ๐ด+๐›ผ๐ฎ๐‘–๐ฏโˆ—๐‘– and ๐ด+๐›ผ๐ฎ๐‘๐ฏโˆ—๐‘ž, ๐‘โ‰ ๐‘ž, where ๐›ผโˆˆโ„, ๐ด is an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}, and ๐ฎ๐‘—,๐ฏ๐‘˜,๐‘—=1,โ€ฆ,๐‘š;๐‘˜=1,โ€ฆ,๐‘›, are the left and right singular vectors, respectively. In particular we give conditions under which this kind of perturbations preserve nonnegativity and certain matrix structures.

1. Introduction

A singular value decomposition of a matrix ๐ดโˆˆโ„‚๐‘šร—๐‘› is a factorization ๐ด=๐‘ˆฮฃ๐‘‰โˆ—, where ฮฃ=diag{๐œŽ1,๐œŽ2,โ€ฆ,๐œŽ๐‘Ÿ}โˆˆโ„๐‘šร—๐‘›,๐‘Ÿ=min{๐‘š,๐‘›},๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿโ‰ฅ0 and both ๐‘ˆโˆˆโ„‚๐‘šร—๐‘š and ๐‘‰โˆˆโ„‚๐‘›ร—๐‘› are unitary. The diagonal entries of ฮฃ are called the singular values of ๐ด. The columns ๐ฎ๐‘— of ๐‘ˆ are called left singular vectors of ๐ด and the columns ๐ฏ๐‘— of ๐‘‰ are called right singular vectors of ๐ด. Every ๐ดโˆˆโ„‚๐‘šร—๐‘› has a singular value decomposition ๐ด=๐‘ˆฮฃ๐‘‰โˆ— and the following relations hold: ๐ด๐ฏ๐‘—=๐œŽ๐‘—๐ฎ๐‘—,๐ดโˆ—๐ฎ๐‘—=๐œŽ๐‘—๐ฏ๐‘—, and ๐ฎโˆ—๐‘—๐ด๐ฏ๐‘—=๐œŽ๐‘—. If ๐ดโˆˆโ„๐‘šร—๐‘›, then ๐‘ˆ and ๐‘‰ may be taken to be real (see [1]).

Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›} and left and right singular vectors ๐ฎ๐‘—,๐ฏ๐‘˜,๐‘—=1,โ€ฆ,๐‘š;๐‘˜=1,โ€ฆ,๐‘›, respectively. In this paper we study how singular values and singular vectors of ๐ด change, under matrix perturbations of the form ๐ด+๐›ผ๐ฎ๐‘–๐ฏโˆ—๐‘– and ๐ด+๐›ผ๐ฎ๐‘๐ฏโˆ—๐‘ž,๐‘โ‰ ๐‘ž,๐›ผโˆˆโ„. Perturbations of the form ๐ด+๐›ผ๐ฎ๐‘–๐ฏโˆ—๐‘– 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 suffcient 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 ๐‘Ÿ eigenvalues of a matrix of order ๐‘›,๐‘Ÿโ‰ค๐‘›, via a perturbation of rank-๐‘Ÿ, without changing any of the ๐‘›โˆ’๐‘Ÿ 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 ๐ด be an ๐‘šร—๐‘› matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿโ‰ฅ0,๐‘Ÿ=min{๐‘š,๐‘›}. Let ๎€ท๐ฎ๐‘ˆ=1โˆฃ๐ฎ2โˆฃโ‹ฏโˆฃ๐ฎ๐‘๎€ธ๎€ท๐ฏ,๐‘‰=1โˆฃ๐ฏ2โˆฃโ‹ฏโˆฃ๐ฏ๐‘๎€ธ,๐‘โ‰ค๐‘Ÿ,(1.1) be matrices of order ๐‘šร—๐‘ and ๐‘›ร—๐‘, whose columns are the left and right singular vectors, respectively, corresponding to ๐œŽ๐‘–,๐‘–=1,โ€ฆ,๐‘. Let ๐ท=diag{๐‘‘1,๐‘‘2,โ€ฆ,๐‘‘๐‘} with ๐œŽ๐‘–+๐‘‘๐‘–โ‰ฅ0. Then ๐ด+UDVโˆ— has singular values ๎€ฝ๐œŽ1+๐‘‘1,โ€ฆ,๐œŽ๐‘+๐‘‘๐‘,๐œŽ๐‘+1,โ€ฆ,๐œŽ๐‘Ÿ๎€พ.(1.2)

Note that the singular values of ๐ด+UDVโˆ— are not necessarily in nondecreasing order. However we can reorderer them by using an appropriate permutation.

Corollary 1.2. Let ๐ด be an ๐‘šร—๐‘› matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿโ‰ฅ0,๐‘Ÿ=min{๐‘š,๐‘›}. Let ๐ฎ๐‘– and ๐ฏ๐‘–, respectively, the left and right singular vectors corresponding to ๐œŽ๐‘–,๐‘–=1,โ€ฆ,๐‘Ÿ. Let ๐›ผโˆˆโ„ such that ๐›ผ+๐œŽ๐‘–โ‰ฅ0,๐‘–=1,โ€ฆ,๐‘Ÿ. Then ๐ด+๐›ผ๐ฎ๐‘–๐ฏโˆ—๐‘– has singular values ๐œŽ1,โ€ฆ,๐œŽ๐‘–โˆ’1,๐œŽ๐‘–+๐›ผ,๐œŽ๐‘–+1,โ€ฆ,๐œŽ๐‘Ÿ.(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 ๐œ…2(๐ด)=๐œŽ1/๐œŽ๐‘Ÿ, then we may choose 0<๐›ผ1โ‰ค๐œŽ1โˆ’๐œŽ2 and 0<๐›ผ๐‘Ÿโ‰ค๐œŽ๐‘Ÿโˆ’1โˆ’๐œŽ๐‘Ÿ in such a way that ๐œ…2๎€ท๐ดโˆ’๐›ผ1๐ฎ1๐ฏโˆ—1+๐›ผ๐‘Ÿ๐ฎ๐‘Ÿ๐ฏโˆ—๐‘Ÿ๎€ธ=๐œŽ1โˆ’๐›ผ1๐œŽ๐‘Ÿ+๐›ผ๐‘Ÿ<๐œ…2(๐ด).(1.4)

The paper is organized as follows. In Section 2 we consider perturbations of the form ๐ด+๐›ผ๐ฎ๐‘–๐ฏโˆ—๐‘–, which we will call simple perturbations, and give sufficient conditions under which the perturbation preserves nonnegativity. In Section 3 we discuss perturbations of the form ๐ด+๐›ผ๐ฎ๐‘๐ฏโˆ—๐‘ž,๐‘โ‰ ๐‘ž, which, because of their different indices, we will call mixed perturbations. We also give sufficient conditions in order that mixed perturbations preserve nonnegativity. It is also shown that both, simple and mixed perturbations, preserve doubly stochastic structure. Finally, we show some examples to illustrate the results.

2. Nonnegativity Preservation under Simple Perturbations

Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}. In this section we consider perturbations ๐ด+๐›ผ๐ฎ๐‘–๐ฏ๐‘‡๐‘–, which preserve nonnegativity. Note that if ๐ด is an ๐‘šร—๐‘› nonnegative matrix, then the left and right singular vectors ๐ฎ1 and ๐ฏ1, corresponding to the maximal singular value ๐œŽ1,respectively, are nonnegative. Hence, in this case, the matrix ๐ด+๐›ผ๐ฎ1๐ฏ๐‘‡1 is nonnegative for all ๐›ผ>0.

Now, let us consider the perturbation ๐ด+๐›ผ๐ฎ๐‘–๐ฏ๐‘‡๐‘–, with ๐‘–>1. Let ๐ฎ๐‘  and ๐ฏ๐‘  be the left and right singular vectors corresponding to ๐œŽ๐‘ ,๐‘ >1, respectively. Let ๐›ผ>0 and let the entry in position (๐‘–,๐‘˜) of ๐ฎ๐‘ ๐ฏ๐‘‡๐‘  be negative. That is, (๐ฎ๐‘ ๐ฏ๐‘‡๐‘ )๐‘–๐‘˜<0. Then if ๐ด=(๐‘Ž๐‘–๐‘˜) is nonnegative, ๐‘Ž๐‘–๐‘˜๎€ท๐ฎ+๐›ผ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜๐‘Žโ‰ฅ0i๏ฌ€0<๐›ผโ‰ค๐‘–๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||.(2.1) Thus, to preserve the nonnegativity of ๐ด it is enough to choose ๐›ผ in the interval ๎ƒฉ0,min๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||๎ƒญ,(2.2) provided that ๐‘Ž๐‘–๐‘˜>0, otherwise (๐ฎ๐‘ ๐ฏ๐‘‡๐‘ )๐‘–๐‘˜ must be zero. Then from (2.2) and Theorem 1.1 we have the following result.

Lemma 2.1. Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}. Let ๐›ผ be in the interval ๎ƒฉ0,min๐‘ min๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||๎ƒญ.(2.3) Then ๐ด+๐›ผ๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ,๐‘ =2,โ€ฆ,๐‘Ÿ, is nonnegative with singular values ๐œŽ1,โ€ฆ,๐œŽ๐‘ โˆ’1,๐œŽ๐‘ +๐›ผ,๐œŽ๐‘ +1,โ€ฆ,๐œŽ๐‘Ÿ.(2.4)

Remark 2.2. It is clear that if in Lemma 2.1, ๐›ผ is taken in ๎ƒฉ0,min๐‘ min๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||๎ƒช,(2.5) then ๐ด+๐›ผ๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ,2โ‰ค๐‘ โ‰ค๐‘Ÿ, is positive with singular values ๐œŽ1,โ€ฆ,๐œŽ๐‘ โˆ’1,๐œŽ๐‘ +๐›ผ,๐œŽ๐‘ +1,โ€ฆ,๐œŽ๐‘Ÿ.(2.6) Moreover, for ๐›ผ in intervals in Lemma 2.1 and this remark, the nonnegativity is obtained independently of the chosen singular vectors ๐ฎ๐‘ ,๐ฏ๐‘ .

A more handle interval for ๐›ผ is given by the following lemma.

Lemma 2.3. Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}. Let ๐›ผ be in the interval ๎ƒฉ๐œŽ0,๐‘Ÿmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—๎ƒญ.(2.7) Then ๐ด+๐›ผ๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ,2โ‰ค๐‘ โ‰ค๐‘Ÿ, is nonnegative with singular values ๐œŽ1,โ€ฆ,๐œŽ๐‘ โˆ’1,๐œŽ๐‘ +๐›ผ,๐œŽ๐‘ +1,โ€ฆ,๐œŽ๐‘Ÿ.(2.8)

Proof. From (2.2) and since |๐‘Ž๐‘–๐‘˜|โ‰ค๐œŽ1, see [1, Corollaryโ€‰โ€‰3.1.3], we have max๐‘–,๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||=max๐‘–,๐‘˜|||||๐‘Ž๐‘–๐‘˜๐œŽ๐‘ โˆ’๐‘Ÿ๎“๐‘—=1,๐‘—โ‰ ๐‘ ๐œŽ๐‘—๐œŽ๐‘ ๎€ท๐ฎ๐‘—๐ฏ๐‘‡๐‘—๎€ธ๐‘–๐‘˜|||||โ‰ค1๐œŽ๐‘ max๐‘–,๐‘˜||๐‘Ž๐‘–๐‘˜||+๐‘Ÿ๎“๐‘—=1,๐‘—โ‰ ๐‘ max๐‘–,๐‘˜๐œŽ๐‘—๐œŽ๐‘ ||๎€ท๐ฎ๐‘—๐ฏ๐‘‡๐‘—๎€ธ๐‘–๐‘˜||โ‰ค1๐œŽ๐‘ max๐‘–,๐‘˜||๐‘Ž๐‘–๐‘˜||+๐‘Ÿ๎“๐‘—=1,๐‘—โ‰ ๐‘ max๐‘–,๐‘˜๐œŽ๐‘—๐œŽ๐‘ โ‰ค๐œŽ1๐œŽ๐‘ +๐‘Ÿ๎“๐‘—=1,๐‘—โ‰ ๐‘ ๐œŽ๐‘—๐œŽ๐‘ โ‰ค1๐œŽ๐‘ ๎ƒฉ2๐œŽ1+๐‘Ÿ๎“๐‘—=2,๐‘—โ‰ ๐‘ ๐œŽ๐‘—๎ƒช=1๐œŽ๐‘ ๎ƒฉ2๐œŽ1โˆ’๐œŽ๐‘ +๐‘Ÿ๎“๐‘—=2๐œŽ๐‘—๎ƒช๐œŽ=21๐œŽ๐‘ โˆ’1+๐‘Ÿ๎“๐‘—=2๐œŽ๐‘—๐œŽ๐‘ ๐œŽโ‰ค21๐œŽ๐‘Ÿโˆ’1+๐‘Ÿ๎“๐‘—=2๐œŽ๐‘—๐œŽ๐‘Ÿ=๐œŽ1๐œŽ๐‘Ÿ+๐‘Ÿโˆ’1๎“๐‘—=1๐œŽ๐‘—๐œŽ๐‘Ÿ.(2.9) Then 1max๐‘–,๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||โ‰ฅ๐œŽ๐‘Ÿ๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—,min๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜max๐‘–,๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||โ‰ฅ๐œŽ๐‘Ÿmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—.(2.10) Hence we have ๎ƒฉ๐œŽ0,๐‘Ÿmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—๎ƒญโŠ†๎ƒฉ0,min๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜max๐‘–,๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||๎ƒญโŠ†๎ƒฉ0,min๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||๎ƒญ,๎ƒฉ๐œŽ0,๐‘Ÿmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—๎ƒญโŠ†๎ƒฉ0,min๐‘ min๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜||๎ƒญ.(2.11) Then, from Lemma 2.1 the result follows.

Remark 2.4. For positive ๐ด=(๐‘Ž๐‘–๐‘—), and ๐›ผโˆˆโ„, we repeat the arguments from Lemmas 2.1 and 2.3 to obtain that if ๐›ผ is in the interval ๎ƒฌโˆ’๐œŽ๐‘Ÿmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—,๐œŽ๐‘Ÿmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—๎ƒญ,(2.12) then ๐ด+๐›ผ๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ,๐‘ =2,โ€ฆ,๐‘Ÿ, is nonnegative with singular values ๐œŽ1,โ€ฆ,๐œŽ๐‘ โˆ’1,||๐œŽ๐‘ ||+๐›ผ,๐œŽ๐‘ +1,โ€ฆ,๐œŽ๐‘Ÿ.(2.13)

Now we consider rank-2 perturbations, ๐ด+UDVโˆ—, where ๐‘ˆ=(๐ฎ๐‘ ,๐ฎ๐‘ก),๐ท=diag{๐›ผ1,๐›ผ2},๐‘‰=(๐ฏ๐‘ ,๐ฏ๐‘ก). That is, perturbations of the form ๐ด+๐›ผ1๐ฎ๐‘ ๐ฏ๐‘‡๐‘ +๐›ผ2๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก as in Theorem 1.1. Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}. Then ๐ด+๐›ผ1๐ฎ๐‘ ๐ฏ๐‘‡๐‘ +๐›ผ2๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก will be nonnegative if ๐‘Ž๐‘–๐‘˜+๐›ผ1๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜+๐›ผ2๎€ท๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก๎€ธ๐‘–๐‘˜โ‰ฅ0,๐‘–=1,โ€ฆ,๐‘š;๐‘—=1,โ€ฆ,๐‘›.(2.14) From the family of straight lines ๐›ผ2๎€ท๐ฎ=โˆ’๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜๎€ท๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก๎€ธ๐‘–๐‘˜๐›ผ1โˆ’๐‘Ž๐‘–๐‘˜๎€ท๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก๎€ธ๐‘–๐‘˜,๐‘–=1,โ€ฆ,๐‘š;๐‘—=1,โ€ฆ,๐‘›,(2.15) it follows that they intersect the axes ๐›ผ1 and ๐›ผ2 at points ๎ƒฉโˆ’๐‘Ž๐‘–๐‘˜๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜๎ƒช,๎ƒฉ๐‘Ž,00,๐‘–๐‘˜๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜๎ƒช,(2.16) where (๐ฎ๐‘ ๐ฏ๐‘‡๐‘ )๐‘–๐‘˜โ‰ 0 and (๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก)๐‘–๐‘˜โ‰ 0, respectively. Let ๐ธ=max๎€ท๐ฎ๐‘–,๐‘˜๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜>0๎ƒฉโˆ’๐‘Ž๐‘–๐‘˜๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜๎ƒช,๐น=max๎€ท๐ฎ๐‘–,๐‘˜๐‘ก๐ฏ๐‘‡๐‘ก๎€ธ๐‘–๐‘˜>0๎ƒฉโˆ’๐‘Ž๐‘–๐‘˜๎€ท๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก๎€ธ๐‘–๐‘˜๎ƒช,๐บ=min๎€ท๐ฎ๐‘–,๐‘˜๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜<0๎ƒฉโˆ’๐‘Ž๐‘–๐‘˜๎€ท๐ฎ๐‘ ๐ฏ๐‘‡๐‘ ๎€ธ๐‘–๐‘˜๎ƒช,๐ป=min๎€ท๐ฎ๐‘–,๐‘˜๐‘ก๐ฏ๐‘‡๐‘ก๎€ธ๐‘–๐‘˜<0๎ƒฉโˆ’๐‘Ž๐‘–๐‘˜๎€ท๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก๎€ธ๐‘–๐‘˜๎ƒช.(2.17) Let ๐‘…1=๎‚†๎€ท๐›ผ1,๐›ผ2๎€ธโˆถ๐ธโ‰ค๐›ผ1๐นโ‰ค0โˆงโˆ’๐ธ๐›ผ1+๐นโ‰ค๐›ผ2๐ปโ‰คโˆ’๐ธ๐›ผ1๎‚‡,๐‘…+๐ป2=๎‚†๎€ท๐›ผ1,๐›ผ2๎€ธโˆถ0โ‰ค๐›ผ1๐นโ‰ค๐บโˆงโˆ’๐บ๐›ผ1+๐นโ‰ค๐›ผ2๐ปโ‰คโˆ’๐บ๐›ผ1๎‚‡.+๐ป(2.18) Then ๐ด+๐›ผ1๐ฎ๐‘ ๐ฏ๐‘‡๐‘ +๐›ผ2๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก is nonnegative for (๐›ผ1,๐›ผ2)โˆˆ๐‘…1โˆช๐‘…2.

A more handle region for (๐›ผ1,๐›ผ2) is given by the following lemma.

Lemma 2.5. Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}. Let ๐‘ƒ1=(๐ธ,0),๐‘ƒ2=(0,๐น),๐‘ƒ3=(๐บ,0),๐‘ƒ4=(0,๐ป) be the intersection points in (2.17). Let ๎€ท๐›ผ1,๐›ผ2๎€ธ๎€ฝโˆˆ๐‘†=(๐‘ฅ,๐‘ฆ)โˆถโ€–(๐‘ฅ,๐‘ฆ)โ€–1๎€พโ‰คmin๐‘˜=1,2,3,4๎€ฝโ€–โ€–๐‘ƒ๐‘˜โ€–โ€–1๎€พ,(2.19) where โ€–โ‹…โ€–1 is the ๐‘™1 norm. Then ๐ด+๐›ผ1๐ฎ๐‘ ๐ฏ๐‘‡๐‘ +๐›ผ2๐ฎ๐‘ก๐ฏ๐‘‡๐‘ก,2โ‰ค๐‘ ,๐‘กโ‰ค๐‘Ÿ, is nonnegative with singular values ๐œŽ1,โ€ฆ,|๐œŽ๐‘ +๐›ผ1|,โ€ฆ,|๐œŽ๐‘ก+๐›ผ2|,โ€ฆ,๐œŽ๐‘Ÿ.

Example 2.6. Let ๎ƒฉ๎ƒช๐ด=124568494,(2.20) with singular values ๐œŽ1=15.5687298,๐œŽ2=3.9581084, and ๐œŽ3=0.9736668. Let ๎€ท๐ฎ๐‘ˆ=2โˆฃ๐ฎ3๎€ธ๎€ท๐ฏ,๐‘‰=2โˆฃ๐ฏ3๎€ธ๎€ฝ๐›ผ,๐ท=diag1,๐›ผ2๎€พ,(2.21) where ๐ฎ2=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 0.42198230.5264618โˆ’0.7380847,๐ฎ3=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ,๐ฏ0.8658718โˆ’0.47531920.15600542=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 0.0257579โˆ’0.66699210.7446195,๐ฏ3=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ .โˆ’0.91068400.29155410.2926616(2.22) From (2.17) we compute ๐ธ,๐น,๐บ,๐ป. Then, the intersection points are (โˆ’12.7300876,0),(7.1058357,0),(0,โˆ’7.9224079),(0,1.2681734),(2.23) and ๐‘†={(๐‘ฅ,๐‘ฆ)โˆถโ€–(๐‘ฅ,๐‘ฆ)โ€–1โ‰ค1.2681734}. Thus, from Lemma 2.5, for (๐›ผ1,๐›ผ2)=(โˆ’1/2,1/2) we have ๐ด+๐›ผ1๐ฎ2๐ฏ๐‘‡2+๐›ผ2๐ฎ3๐ฏ๐‘‡3=๎ƒฉ๎ƒช0.600302.26703.96965.20976.10637.73443.93858.77664.2976,(2.24) with singular values ๐œŽ1,๐œŽ2+๐›ผ1,๐œŽ3+๐›ผ2.

Remark 2.7. Let ๐ด be an ๐‘šร—๐‘› complex matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›} and singular value decomposition ๐ด=๐‘ˆฮฃ๐‘‰โˆ—. In [11] it was defined the concept of energy of ๐ด as โ„ฐ(๐ด)=๐œŽ1+๐œŽ2+โ‹ฏ+๐œŽ๐‘Ÿ. If ๐ด is positive, then as an application of the rank-2 perturbation result, from Lemma 2.5 and Remark 2.4, we may construct, for ๎ƒฏ๐œŽ0<๐›ผโ‰คmin๐‘Ÿmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜๐œŽ1+โˆ‘๐‘Ÿโˆ’1๐‘—=1๐œŽ๐‘—,๐œŽ๐‘—๎ƒฐ,(2.25) a family of nonnegative matrices ๐ต=๐ด+๐›ผ๐ฎ๐‘–๐ฏ๐‘‡๐‘–โˆ’๐›ผ๐ฎ๐‘—๐ฏ๐‘‡๐‘—, with โ„ฐ(๐ต)=โ„ฐ(๐ด). Now, suppose ๐ด is nonnegative with โ„ฐ(๐ด)โ‰ค๐ถ, where ๐ถ is an upper bound. Then from (1.2), by taking ๐›ผ=๐ถโˆ’โ„ฐ(๐ด) we may construct a family of nonnegative matrices ๐ต=๐ด+๐›ผ๐ฎ1๐ฏ๐‘‡1 with โ„ฐ(๐ต)=โ„ฐ(๐ด)+๐›ผ=๐ถ.

Now, in order to show that simple perturbations preserve doubly stochastic structure we need the following lemma. First we introduce a definition and a notation. An ๐‘›ร—๐‘› matrix ๐ด=(๐‘Ž๐‘–๐‘—) is said to be with constant row sums if โˆ‘๐‘›๐‘—=1๐‘Ž๐‘–๐‘—=๐›ผ,๐‘–=1,2,โ€ฆ,๐‘›. We denote by CS๐›ผ the set of all matrices with constant row sums equal to ๐›ผ.

Lemma 2.8. Let ๐ด be an ๐‘›ร—๐‘› irreducible doubly stochastic matrix and let ๐ด๐ฑ=๐œ†๐ฑ,๐ฑ๐‘‡=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›), with ๐œ†โ‰ 1. Then ๐‘†(๐ฑ)=๐‘ฅ1+๐‘ฅ2+โ‹ฏ+๐‘ฅ๐‘›=0.(2.26)

Proof. Since ๐ด is doubly stochastic, then ๐‘†(๐ด๐ฑ)=ฮฃ๐‘Ž1๐‘—๐‘ฅ๐‘—+ฮฃ๐‘Ž2๐‘—๐‘ฅ๐‘—+โ‹ฏ+ฮฃ๐‘Ž๐‘›๐‘—๐‘ฅ๐‘—=๐‘ฅ1ฮฃ๐‘Ž๐‘–1+๐‘ฅ2ฮฃ๐‘Ž๐‘–2+โ‹ฏ+๐‘ฅ๐‘›ฮฃ๐‘Ž๐‘–๐‘›=๐‘†(๐‘ฅ),(2.27) and ๐‘†(๐ด๐ฑ)=๐‘†(๐œ†๐ฑ)=๐œ†๐‘†(๐ฑ). Then, ๐‘†(๐ฑ)=๐œ†๐‘†(๐ฑ),(2.28) and since ๐œ†โ‰ 1,๐‘†(๐ฑ)=0.

The following result shows that simple perturbations preserve doubly stochastic structure.

Proposition 2.9. Let ๐ด be an ๐‘›ร—๐‘› irreducible doubly stochastic matrix. Then, ๎€ท(i)๐ด+๐›ผ1๐ฎ1๐ฏ๐‘‡1๎€ธโˆˆCS1+๐›ผ1,๎€ท๐ด+๐›ผ1๐ฎ1๐ฏ๐‘‡1๎€ธ๐‘‡โˆˆCS1+๐›ผ1,๎€ท(ii)๐ด+๐›ผ๐‘–๐ฎ๐‘–๐ฏ๐‘‡๐‘–๎€ธโˆˆCS1,๎€ท๐ด+๐›ผ๐‘–๐ฎ๐‘–๐ฏ๐‘‡๐‘–๎€ธ๐‘‡โˆˆCS1๎ƒฉ;๐‘–=2,โ€ฆ,๐‘›,(iii)๐ด+๐‘›๎“๐‘–=1๐›ผ๐‘–๐ฎ๐‘–๐ฏ๐‘‡๐‘–๎ƒชโˆˆCS1+๐›ผ1.(2.29)

Proof. Since ๐ด,๐ด๐‘‡โˆˆCS1, then ๐ด๐ด๐‘‡โˆˆCS1 and ๐ด๐‘‡๐ดโˆˆCS1. Hence, the singular vectors ๐ฎ1 and ๐ฏ1 are ๐ฎ1=๐ฏ1=1โˆš๐‘›1๐ž=โˆš๐‘›(1,1,โ€ฆ,1)๐‘‡.(2.30) Then ๐›ผ1๐ฎ1๐ฏ๐‘‡1=๐›ผ1๐ฏ1๐ฎ๐‘‡1=(1/๐‘›)๐›ผ1๐ž๐ž๐‘‡ and ๐›ผ1๐ฎ1๐ฏ๐‘‡1โˆˆCS๐›ผ1. Thus (i) holds. From Lemma 2.8, ๐›ผ๐‘–๐ฎ๐‘–๐ฏ๐‘‡๐‘–๐ž=๐›ผ๐‘–(๐ฏ๐‘‡๐‘–๐ž)๐ฎ๐‘–=0 and (ii) holds. From (i) and (ii) we have (iii).

Example 2.10. Consider the matrix โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๐ด=4321143221433214,(2.31) which is nonnegative generalized doubly stochastic, that is, ๐ด is nonnegative with ๐ด,๐ด๐‘‡โˆˆCS10, and ๐ด has singular values 10,2.8284,2,8284,2. Let ๐‘ˆฮฃ๐‘‰โˆ— be the singular value decomposition of ๐ด. Let ๐ท=diag{4,3,2,1}. Then ๐ต=๐ด+๐‘ˆ๐ท๐‘‰โˆ—=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 6.25964.85002.24040.65011.08226.00814.41782.49192.24040.65016.25964.85004.41782.49191.08226.0081(2.32) is nonnegative generalized doubly stochastic with singular values 14,5.8284,4.8284,3.

3. Nonnegativity Preservation under Mixed Perturbations

In this section we discuss matrix perturbations of the form ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘—, with ฬ‡๐‘˜โ‰ ๐‘—, which we will call mixed perturbations, and we study how the singular values and vectors change under this kind of perturbations. We also give sufficient conditions under which mixed perturbations preserve nonnegativity and preserve doubly stochastic structure. Let us start by considering the following particular case: let ๐ด be a 4ร—3 matrix with singular values ๐œŽ1,๐œŽ2,๐œŽ3. Let ๐ด=๐‘ˆฮฃ๐‘‰โˆ— with ๐‘ˆ=(๐ฎ1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4) and๐‘‰=(๐ฏ1โˆฃ๐ฏ2โˆฃ๐ฏ3). Let ๐›ผโ‰ฅ0. That is, ๎€ท๐ฎ๐ด=1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4๎€ธโŽ›โŽœโŽœโŽ๐œŽ1000๐œŽ2000๐œŽ3โŽžโŽŸโŽŸโŽ ๎ƒฉ๐ฏ000โˆ—1๐ฏโˆ—2๐ฏโˆ—3๎ƒช=3๎“๐‘˜=1๐œŽ๐‘˜๐ฎ๐‘˜๐ฏโˆ—๐‘˜.(3.1) The matrix ๐›ผ๐ฎ1๐ฏโˆ—2 has the singular value decomposition: ๐›ผ๐ฎ1๐ฏโˆ—2=๎€ท๐ฎ1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4๎€ธโŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๎ƒฉ๐ฏ๐›ผ00000000000โˆ—2๐ฏโˆ—1๐ฏโˆ—3๎ƒช=๎€ท๐ฎ1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4๎€ธโŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๎ƒฉ๐ฏ๐›ผ00000000000010100001๎ƒช๎ƒฉโˆ—1๐ฏโˆ—2๐ฏโˆ—3๎ƒช=๎€ท๐ฎ1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4๎€ธโŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๎ƒฉ๐ฏ0๐›ผ0000000000โˆ—1๐ฏโˆ—2๐ฏโˆ—3๎ƒช.(3.2) Then, ๐ด+๐›ผ๐ฎ1๐ฏโˆ—2=๎€ท๐ฎ1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4๎€ธโŽ›โŽœโŽœโŽ๐œŽ1๐›ผ00๐œŽ2000๐œŽ3โŽžโŽŸโŽŸโŽ ๎ƒฉ๐ฏ000โˆ—1๐ฏโˆ—2๐ฏโˆ—3๎ƒช.(3.3) Now we compute the singular values of the matrix โŽ›โŽœโŽœโŽ๐œŽ๐ถ=1๐›ผ00๐œŽ2000๐œŽ3โŽžโŽŸโŽŸโŽ 000,(3.4) by computing the eigenvalues of ๎‚๐ถ๎‚๐ถ๐‘‡, where ๎‚๐ถ=(๐œŽ1๐›ผ0๐œŽ2). Since ๎‚๐ถ๎‚๐ถ๐‘‡=๎‚ต๐›ผ2+๐œŽ21๐›ผ๐œŽ2๐›ผ๐œŽ2๐œŽ22๎‚ถ,(3.5) then ๎‚€๎‚๐ถ๎‚๐ถtr๐‘‡๎‚=๐›ผ2+๐œŽ21+๐œŽ22=๐œ†1+๐œ†2,๎‚€๎‚๐ถ๎‚๐ถdet๐‘‡๎‚=๐œŽ21๐œŽ22=๐œ†1๐œ†2(3.6) with ๐œ†1,๐œ†2 being the eigenvalues of ๎‚๐ถ๎‚๐ถโˆ—. Thus, we obtain ๐œ†1=๐›ผ2+๐œŽ21+๐œŽ22+๎”๎€ท๐›ผ2+๐œŽ21+๐œŽ22๎€ธ2โˆ’4๐œŽ21๐œŽ222,๐œ†2=๐›ผ2+๐œŽ21+๐œŽ22โˆ’๎”๎€ท๐›ผ2+๐œŽ21+๐œŽ22๎€ธ2โˆ’4๐œŽ21๐œŽ222.(3.7) Hence, the singular values of ๐ด+๐›ผ๐ฎ1๐ฏโˆ—2 are โˆš๐œ†1,โˆš๐œ†2,๐œŽ3.

Now we generalize these results for ๐‘šร—๐‘› matrices.

Theorem 3.1. Let ๐ด be an ๐‘šร—๐‘› matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿโ‰ฅ0,๐‘Ÿ=min{๐‘š,๐‘›}, and with singular value decomposition ๐ด=๐‘ˆฮฃ๐‘‰โˆ—, where ๎€ท๐ฎ๐‘ˆ=1โˆฃโ‹ฏโˆฃ๐ฎ๐‘˜โˆฃโ‹ฏโˆฃ๐ฎ๐‘š๎€ธ๎€ท๐ฏ,๐‘‰=1โˆฃโ‹ฏโˆฃ๐ฏ๐‘—โˆฃโ‹ฏโˆฃ๐ฏ๐‘›๎€ธ,๎€ท๐œŽฮฃ=diag1,๐œŽ2,โ€ฆ,๐œŽ๐‘Ÿ๎€ธ,๐‘˜,๐‘—โ‰ค๐‘Ÿ.(3.8) Let ๐›ผโˆˆโ„. Then ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘— has singular values, ๎€ฝ๐œŽ1,โ€ฆ๐œŽ๐‘˜โˆ’1,๎‚๐œŽ๐‘˜,๐œŽ๐‘˜+1,โ€ฆ,๐œŽ๐‘—โˆ’1,๎‚๐œŽ๐‘—,๐œŽ๐‘—+1,โ€ฆ,๐œŽ๐‘Ÿ๎€พ,(3.9) where ๎‚๐œŽ๐‘˜=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ2+๐œŽ2๐‘˜+๐œŽ2๐‘—+๎‚™๎‚€๐›ผ2+๐œŽ2๐‘˜+๐œŽ2๐‘—๎‚2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—2โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2,(3.10)๎‚๐œŽ๐‘—=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ2+๐œŽ2๐‘˜+๐œŽ2๐‘—โˆ’๎‚™๎‚€๐›ผ2+๐œŽ2๐‘˜+๐œŽ2๐‘—๎‚2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—2โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2.(3.11)

Proof. Without loss of generality we may assume that ๐‘šโ‰ฅ๐‘› and ๐‘—>๐‘˜. The matrix ๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘— has a singular value decomposition ๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘—=๎€ทยฑ๐ฎ๐‘˜โˆฃโ‹ฏโˆฃ๐ฎ๐‘˜โˆ’1โˆฃ๐ฎ1โˆฃ๐ฎ๐‘˜+1โˆฃโ‹ฏโˆฃ๐ฎ๐‘š๎€ธ๎‚ฮฃโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐ฏโˆ—๐‘—โ‹ฎ๐ฏโˆ—๐‘—โˆ’1๐ฏโˆ—1๐ฏโˆ—๐‘—+1โ‹ฎ๐ฏโˆ—๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(3.12) where ๎‚ฮฃ=diag{|๐›ผ|,0,โ€ฆ,0}. The decomposition in (3.12) can be written as ๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘—=๐‘ˆ๐‘ƒ๐‘˜๎‚ฮฃ๐‘„๐‘—๐‘‰โˆ—,(3.13) where ๐‘ƒ๐‘˜ and ๐‘„๐‘— are ๐‘šร—๐‘š and ๐‘›ร—๐‘› permutation matrices of the form ๐‘ƒ๐‘˜=๎‚€๐‘Š๐‘˜00๐ผ๐‘šโˆ’๐‘˜๎‚,๐‘„๐‘—=๎‚ต๐‘Š๐‘—00๐ผ๐‘›โˆ’๐‘—๎‚ถ,(3.14) with ๐‘Š๐‘™=๎ƒฉ1๐ผ๐‘™โˆ’21๎ƒชoforder๐‘™.(3.15) Since ๎‚๎ƒฉ๐›ผโ‹ฑโ‹ฑ๎ƒชฮฃ=,then๐‘ƒ๐‘˜๎‚ฮฃ๐‘„๐‘—=๎ƒฉโ‹ฑ(๐›ผ)๐‘˜,๐‘—โ‹ฑ๎ƒช,(3.16) where if ๐›ผ<0, we multiply the ๐ฎ๐‘˜ vector by minus one. Then ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘—๎‚€=๐‘ˆฮฃ+๐‘ƒ๐‘˜๎‚ฮฃ๐‘„๐‘—๎‚๐‘‰โˆ—๎‚€ฮ›0๎‚๐‘‰=๐‘ˆโˆ—,(3.17) where โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐œŽฮ›=1โ‹ฑ๐œŽ๐‘˜๐œŽโ‹ฏ๐›ผโ‹ฑโ‹ฎ๐‘—๐œŽ๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(3.18) By applying row and column permutations on the matrix ฮฃ+๐‘ƒ๐‘˜๎‚ฮฃ๐‘„๐‘—, it follows from (3.17) and (3.18) that the singular values of ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘— are ๎€ฝ๐œŽ1,โ€ฆ,๐œŽ๐‘˜โˆ’1,๎‚๐œŽ๐‘˜,๐œŽ๐‘˜+1,โ€ฆ,๐œŽ๐‘—โˆ’1,๎‚๐œŽ๐‘—,๐œŽ๐‘—+1,โ€ฆ,๐œŽ๐‘›๎€พ,(3.19) where ๎‚๐œŽ๐‘˜ and ๎‚๐œŽ๐‘— are as in (3.10) and (3.11), respectively.

Observe that in Theorem 3.1, ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘— has singular values ๎€ฝ๐œŽ1,โ€ฆ๐œŽ๐‘˜โˆ’1,๎‚๐œŽ๐‘˜,๐œŽ๐‘˜+1,โ€ฆ,๐œŽ๐‘—โˆ’1,๎‚๐œŽ๐‘—,๐œŽ๐‘—+1,โ€ฆ,๐œŽ๐‘Ÿ๎€พ,(3.20) if ๐‘˜,๐‘—โ‰ค๐‘Ÿ. If ๐‘Ÿ<๐‘˜โ‰ค๐‘š, then only ๐œŽ๐‘—, corresponding to the right singular vector ๐ฏ๐‘—, changes and take the form ๎‚๐œŽ๐‘—=๎”๐›ผ2+๐œŽ2๐‘—. A straightforward calculation shows that for ๐›ผ>0,๎‚๐œŽ๐‘˜โ‰ค๐œŽ๐‘˜+๐›ผ. Observe that ๎‚๐œŽ๐‘—=๎”๐›ผ2+๐œŽ2๐‘—โ‰ค๐œŽ๐‘—+๐›ผ.(3.21)

Example 3.2. Let โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๐ด=120613041150,(3.22) with singular values ๐œŽ1=7.8207,๐œŽ2=5.6257,๐œŽ3=1.09.Let ๐ฎ1=๎€ท๎€ธ0.260300.703860.397760.52784๐‘‡,๐ฏ2=๎€ท๎€ธ0.63252โˆ’0.716470.29425๐‘‡,(3.23) left and right singular vectors of ๐ด, respectively. Let ๐›ผ=1. Then, from (3.10) and (3.11), the matrix ๐ด+๐›ผ๐ฎ1๐ฏโˆ—2=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 1.16461.81350.0765936.44520.495713.20710.251593.7151.1171.33394.62180.15532(3.24) has singular values ๎‚๐œŽ1=7.9477,๎‚๐œŽ2=5.5358 and ๐œŽ3. By using Theorem 1.1 we have that ๐ด+๐ฎ1๐ฏโˆ—1+๐ฎ2๐ฏโˆ—2 has singular values ๐œŽ1+1=8.8207,๐œŽ2+1=6.6257,and๐œŽ3=1.09.

Different from perturbations of the form ๐ด+๐›ผ๐ฎ๐‘–๐ฏ๐‘‡๐‘–, the perturbation of Theorem 3.1 affects not only the singular values ๐œŽ๐‘˜ and ๐œŽ๐‘—, but also the corresponding left and right singular vectors ๐ฎ๐‘˜ and ๐ฏ๐‘—. To make this modification clear, we consider again the previous discussion to Theorem 3.1: ๐ด+๐›ผ๐ฎ1๐ฏโˆ—2=๎€ท๐ฎ1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4๎€ธโŽ›โŽœโŽœโŽ๐œŽ1๐›ผ00๐œŽ2000๐œŽ3โŽžโŽŸโŽŸโŽ ๎ƒฉ๐ฏ000โˆ—1๐ฏโˆ—2๐ฏโˆ—3๎ƒช.(3.25) Let ๎‚๎‚€๐œŽ๐ถ=1๐›ผ0๐œŽ2๎‚=๎‚€๐œ”11๐œ”12๐œ”21๐œ”22๎‚๎‚ต๎‚๐œŽ100๎‚๐œŽ2๎‚ถ๎‚€๐œ11๐œ12๐œ21๐œ22๎‚๐‘‡,(3.26) the SVD of ๎‚๐ถ with ๎‚๐œŽ1,๎‚๐œŽ2 obtained from (3.10) and (3.11), respectively. The left singular vectors of โŽ›โŽœโŽœโŽ๐œŽ๐ถ=1๐›ผ00๐œŽ2000๐œŽ3โŽžโŽŸโŽŸโŽ 000(3.27) (eigenvectors of ๐ถ๐ถ๐‘‡) are ฬƒ๐ฎ1=๎€ท๐œ”11๐œ”21๎€ธ00๐‘‡,ฬƒ๐ฎ2=๎€ท๐œ”12๐œ”22๎€ธ00๐‘‡,ฬƒ๐ฎ3=๎€ท๎€ธ0010๐‘‡=๐‘’3,ฬƒ๐ฎ4=๎€ท๎€ธ0001๐‘‡=๐‘’4,(3.28) and its corresponding right singular vectors (eigenvectors of ๐ถ๐‘‡๐ถ) are ฬƒ๐ฏ1=๎€ท๐œ11๐œ210๎€ธ๐‘‡,ฬƒ๐ฏ2=๎€ท๐œ12๐œ220๎€ธ๐‘‡,ฬƒ๐ฏ3=๎€ท๎€ธ001๐‘‡=๐‘’3.(3.29) Thus, ๐ด+๐›ผ๐ฎ1๐ฏโˆ—2 can be written as ๐ด+๐›ผ๐ฎ1๐ฏโˆ—2=๎‚๐‘ˆ๎‚ฮฃ๎‚๐‘‰โˆ—, where ๎‚๎€ท๐ฎ๐‘ˆ=1โˆฃ๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4ฬƒ๐ฎ๎€ธ๎€ท1โˆฃฬƒ๐ฎ2โˆฃฬƒ๐ฎ3โˆฃฬƒ๐ฎ4๎€ธ=๎€ท๐œ”11๐ฎ1+๐œ”21๐ฎ2โˆฃ๐œ”12๐ฎ1+๐œ”22๐ฎ2โˆฃ๐ฎ3โˆฃ๐ฎ4๎€ธ,๎‚๎€ท๐ฏ๐‘‰=1โˆฃ๐ฏ2โˆฃ๐ฏ3ฬƒ๐ฏ๎€ธ๎€ท1โˆฃฬƒ๐ฏ2โˆฃฬƒ๐ฏ3๎€ธ=๎€ท๐œ11๐ฏ1+๐œ21๐ฏ2โˆฃ๐œ12๐ฏ1+๐œ22๐ฏ2โˆฃ๐ฏ3๎€ธ,๎‚๎€ทฮฃ=diag๎‚๐œŽ1,๎‚๐œŽ2,๐œŽ3๎€ธ.(3.30) Now, we generalize this result. Without loss generality we assume that ๐‘˜<๐‘—. From (3.17) and (3.18), by permuting rows and columns of the matrix ฮฃ+๐‘ƒ๐‘˜๎‚ฮฃ๐‘„๐‘—, we have ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘—=๐‘ˆ1๎‚ตฮ›0๎‚ถ๐‘‰โˆ—1โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œŽwithฮ›=1โ‹ฑ๐œŽ๐‘˜๐›ผ๐œŽ๐‘—๐œŽโ‹ฑ0๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘ˆ1=๎€ท๐ฎ1โˆฃโ‹ฏโˆฃ๐ฎ๐‘˜โˆฃ๐ฎ๐‘—โˆฃโ‹ฏโˆฃ๐ฎ๐‘˜+1โˆฃ๐ฎ๐‘—+1โˆฃโ‹ฏโˆฃ๐ฎ๐‘š๎€ธ,๐‘‰1=๎€ท๐ฏ1โˆฃโ‹ฏโˆฃ๐ฏ๐‘˜โˆฃ๐ฏ๐‘—โˆฃโ‹ฏโˆฃ๐ฏ๐‘˜+1โˆฃ๐ฏ๐‘—+1โˆฃโ‹ฏโˆฃ๐ฏ๐‘›๎€ธ.(3.31) The singular vectors of (ฮ›0) are obtained from the singular vectors of the 2ร—2 matrix (๐œŽ๐‘˜๐›ผ0๐œŽ๐‘—). Let ๎‚ต๐œŽ๐‘˜๐›ผ0๐œŽ๐‘—๎‚ถ=๎‚€๐œ”11๐œ”12๐œ”21๐œ”22๎‚๎‚ต๎‚๐œŽ๐‘˜00๎‚๐œŽ๐‘—๎‚ถ๎‚€๐œ11๐œ12๐œ21๐œ22๎‚๐‘‡.(3.32) Then the unitary matrices of the singular value decomposition of (ฮ›0) are ๐‘ˆ2=โŽ›โŽœโŽœโŽ๐ผ๐œ”11๐œ”12๐œ”21๐œ”22๐ผโŽžโŽŸโŽŸโŽ =๎€ท๐ž1โˆฃโ‹ฏโˆฃ๐ž๐‘˜โˆ’1โˆฃฬƒ๐ฎ๐‘˜โˆฃฬƒ๐ฎ๐‘˜+1โˆฃ๐ž๐‘˜+2โˆฃโ‹ฏโˆฃ๐ž๐‘š๎€ธ,๐‘‰2=โŽ›โŽœโŽœโŽ๐ผ๐‘ฃ11๐‘ฃ12๐‘ฃ21๐‘ฃ22๐ผโŽžโŽŸโŽŸโŽ =๎€ท๐ž1โˆฃโ‹ฏโˆฃ๐ž๐‘˜โˆ’1โˆฃฬƒ๐ฏ๐‘˜โˆฃฬƒ๐ฏ๐‘˜+1โˆฃ๐ž๐‘˜+2โˆฃโ‹ฏโˆฃ๐ž๐‘›๎€ธ,(3.33) of order ๐‘šร—๐‘š and ๐‘›ร—๐‘›, respectively. Then ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘—=๎‚๐‘ˆ๎‚ต๎‚ฮ›0๎‚ถ๎‚๐‘‰โˆ—๎‚with๐‘ˆ=๐‘ˆ1๐‘ˆ2,๎‚๐‘‰=๐‘‰1๐‘‰2,๎‚โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œŽฮ›=1โ‹ฑ๎‚๐œŽ๐‘˜๎‚๐œŽ๐‘—โ‹ฑ๐œŽ๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(3.34) where ๎‚๐œŽ๐‘˜ and ๎‚๐œŽ๐‘— are not ordered. Since ๎‚๎€ท๐ฎ๐‘ˆ=1โˆฃโ‹ฏโˆฃ๐œ”11๐ฎ๐‘˜+๐œ”21๐ฎ๐‘—โˆฃ๐œ”12๐ฎ๐‘˜+๐œ”22๐ฎ๐‘—โˆฃโ‹ฏโˆฃ๐ฎ๐‘š๎€ธ,๎‚๎€ท๐ฏ๐‘‰=1โˆฃโ‹ฏโˆฃ๐œ11๐ฏ๐‘˜+๐œ21๐ฏ๐‘—โˆฃ๐œ12๐ฏ๐‘˜+๐œ22๐ฏ๐‘—โˆฃโ‹ฏโˆฃ๐ฏ๐‘›๎€ธ,(3.35) then the singular vectors corresponding to ๐œŽ๐‘˜,๐œŽ๐‘— have been modified. We have prove the following result.

Corollary 3.3. Let ๐ด be an ๐‘šร—๐‘› matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›} and with singular value decomposition ๐ด=๐‘ˆฮฃ๐‘‰โˆ—, where ๎€ท๐ฎ๐‘ˆ=1โˆฃโ‹ฏโˆฃ๐ฎ๐‘˜โˆฃโ‹ฏโˆฃ๐ฎ๐‘š๎€ธ๎€ท๐ฏ,๐‘‰=1โˆฃโ‹ฏโˆฃ๐ฏ๐‘—โˆฃโ‹ฏ๐ฏ๐‘›๎€ธ๎€ฝ๐œŽ,๐‘˜,๐‘—โ‰ค๐‘Ÿ,ฮฃ=diag1,๐œŽ2,โ€ฆ,๐œŽ๐‘Ÿ๎€พ.(3.36) Let ๐›ผโˆˆโ„. Then ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘— has left singular vectors, ฬƒ๐ฎ๐‘–=๐ฎ๐‘–ฬƒ๐ฎ,๐‘–=1,โ€ฆ,๐‘š,๐‘–โ‰ ๐‘˜,๐‘–โ‰ ๐‘—,๐‘˜=๐œ”11๐ฎ๐‘˜+๐œ”21๐ฎ๐‘—,ฬƒ๐ฎ๐‘—=๐œ”12๐ฎ๐‘˜+๐œ”22๐ฎ๐‘—(3.37) and right singular vectors ฬƒ๐ฏ๐‘–=๐ฏ๐‘–ฬƒ๐ฏ,๐‘–=1,โ€ฆ,๐‘›,๐‘–โ‰ ๐‘˜,๐‘–โ‰ ๐‘—,๐‘˜=๐œ11๐ฏ๐‘˜+๐œ21๐ฏ๐‘—,ฬƒ๐ฏ๐‘—=๐œ12๐ฏ๐‘˜+๐œ22๐ฏ๐‘—.(3.38)

Observe that if 2ร—2 orthogonal matrices in (3.32) are of the same type ๎‚€๎‚๎‚€๎‚๐‘๐‘ โˆ’๐‘ ๐‘or๐‘๐‘ ๐‘ โˆ’๐‘,(3.39) then a straight forward calculation shows that (3.37) and (3.38) become ฬƒ๐ฎ๐‘–=๐ฎ๐‘–ฬƒ๐ฎ,๐‘–=1,โ€ฆ,๐‘š,๐‘–โ‰ ๐‘˜,๐‘–โ‰ ๐‘—,๐‘˜=๐‘1๐ฎ๐‘˜โˆ’๐‘ 1๐ฎ๐‘—,ฬƒ๐ฎ๐‘—=๐‘ 1๐ฎ๐‘˜+๐‘1๐ฎ๐‘—(3.40) and ฬƒ๐ฏ๐‘–=๐ฏ๐‘–ฬƒ๐ฏ,๐‘–=1,โ€ฆ,๐‘›,๐‘–โ‰ ๐‘˜,๐‘–โ‰ ๐‘—,๐‘˜=๐‘2๐ฏ๐‘˜โˆ’๐‘ 2๐ฏ๐‘—,ฬƒ๐ฏ๐‘—=๐‘ 2๐ฏ๐‘˜+๐‘2๐ฏ๐‘—,(3.41) respectively, while if they are of different type, then (3.38) becomes ฬƒ๐ฏ๐‘–=๐ฏ๐‘–ฬƒ๐ฏ,๐‘–=1,โ€ฆ,๐‘š,๐‘–โ‰ ๐‘˜,๐‘–โ‰ ๐‘—,๐‘˜=๐‘2๐ฏ๐‘˜+๐‘ 2๐ฏ๐‘—,ฬƒ๐ฏ๐‘—=๐‘ 2๐ฏ๐‘˜โˆ’๐‘2๐ฏ๐‘—.(3.42) From (3.10) and (3.11) it is clear that ๎‚๐œŽ๐‘˜>๎‚๐œŽ๐‘— with ๐‘˜<๐‘—. The following result tells us how the new singular values ๎‚๐œŽ๐‘˜,๎‚๐œŽ๐‘— relate with the previous singular values ๐œŽ๐‘˜,๐œŽ๐‘—.

Corollary 3.4. Let ๐ด be an ๐‘šร—๐‘› matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿโ‰ฅ0,๐‘Ÿ=min{๐‘š,๐‘›} and with singular value decomposition ๐ด=๐‘ˆฮฃ๐‘‰โˆ—, where ๎€ท๐ฎ๐‘ˆ=1โˆฃโ‹ฏโˆฃ๐ฎ๐‘˜โˆฃโ‹ฏโˆฃ๐ฎ๐‘š๎€ธ๎€ท๐ฏ,๐‘‰=1โˆฃโ‹ฏโˆฃ๐ฏ๐‘—โˆฃโ‹ฏ๐ฏ๐‘›๎€ธ๎€ฝ๐œŽ,๐‘˜,๐‘—โ‰ค๐‘Ÿ,ฮฃ=diag1,๐œŽ2,โ€ฆ,๐œŽ๐‘Ÿ๎€พ.(3.43) Let ๎‚๐œŽ๐‘˜,๎‚๐œŽ๐‘—,๐‘˜<๐‘—, be respectively, as in (3.10) and (3.11), the singular values of ๐ด+๐›ผ๐ฎ๐‘˜๐ฏโˆ—๐‘—. Then ๎‚๐œŽ๐‘˜>๐œŽ๐‘˜โ‰ฅ๐œŽ๐‘—>๎‚๐œŽ๐‘—.

Proof. Since 4๐œŽ2๐‘—๐›ผ2>0 we have ๎€ท๐œŽ2๐‘˜+๐›ผ2๎€ธ2+๐œŽ4๐‘—+2๐œŽ2๐‘—๎€ท๐œŽ2๐‘˜+๐›ผ2๎€ธโˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—>๎€ท๐œŽ2๐‘˜+๐›ผ2๎€ธ2+๐œŽ4๐‘—โˆ’2๐œŽ2๐‘—๎€ท๐œŽ2๐‘˜+๐›ผ2๎€ธ,๎€ท๐œŽ2๐‘˜+๐œŽ2๐‘—+๐›ผ2๎€ธ2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—>๐œŽ๎€ท๎€ท2๐‘˜+๐›ผ2๎€ธโˆ’๐œŽ2๐‘—๎€ธ2,๐œŽ2๐‘˜+๐›ผ2+๐œŽ2๐‘—โˆ’๎‚™๎‚€๐œŽ2๐‘˜+๐œŽ2๐‘—+๐›ผ2๎‚2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—<2๐œŽ2๐‘—.(3.44) Thus ๐œŽ๐‘—>๎ƒฉ12๎ƒฌ๐œŽ2๐‘˜+๐›ผ2+๐œŽ2๐‘—โˆ’๎‚™๎‚€๐œŽ2๐‘˜+๐œŽ2๐‘—+๐›ผ2๎‚2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—๎ƒญ๎ƒช1/2.(3.45) In the same way, ๎€ท๐œŽ2๐‘—+๐›ผ2๎€ธ2+๐œŽ4๐‘˜+2๐œŽ2๐‘˜๎€ท๐œŽ2๐‘—+๐›ผ2๎€ธโˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—>๎€ท๐œŽ2๐‘—+๐›ผ2๎€ธ2+๐œŽ4๐‘˜โˆ’2๐œŽ2๐‘˜๎€ท๐œŽ2๐‘—+๐›ผ2๎€ธ,๎€ท๐œŽ2๐‘˜+๐œŽ2๐‘—+๐›ผ2๎€ธ2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—>๎€ท๐œŽ2๐‘˜โˆ’๎€ท๐œŽ2๐‘—+๐›ผ2๎€ธ๎€ธ2,๐œŽ2๐‘˜+๐œŽ2๐‘—+๐›ผ2+๎‚™๎‚€๐œŽ2๐‘˜+๐œŽ2๐‘—+๐›ผ2๎‚2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—>2๐œŽ2๐‘˜.(3.46) Then ๐œŽ๐‘˜<๎ƒฉ12๎ƒฌ๐œŽ2๐‘˜+๐›ผ2+๐œŽ2๐‘—+๎‚™๎‚€๐œŽ2๐‘˜+๐œŽ2๐‘—+๐›ผ2๎‚2โˆ’4๐œŽ2๐‘˜๐œŽ2๐‘—๎ƒญ๎ƒช1/2.(3.47) Therefore, ๎‚๐œŽ๐‘˜>๐œŽ๐‘˜โ‰ฅ๐œŽ๐‘—>๎‚๐œŽ๐‘—.(3.48) Observe that ๐œŽ๐‘–โˆˆ(๎‚๐œŽ๐‘—,๎‚๐œŽ๐‘˜),๐‘–=๐‘˜,๐‘˜+1,โ€ฆ,๐‘—. In particular for ๐‘˜=1 and ๐‘—=๐‘›, all singular values of ๐ด are in the interval (๎‚๐œŽ๐‘›,๎‚๐œŽ1).

Now we extend the mixed perturbation result given by Theorem 3.1 to rank-2 perturbations, that is, perturbations of the form ๐ต=๐ด+๐›ผ1๐ฎ๐‘˜1๐ฏ๐‘—1+๐›ผ2๐ฎ๐‘˜2๐ฏ๐‘—2, with ๐›ผ1,๐›ผ2 nonzero real numbers and ๐ฎ๐‘˜๐‘–,๐ฏ๐‘—๐‘–,๐‘–=1,2,๐‘˜๐‘–<๐‘—๐‘–,๐‘˜1<๐‘—1<๐‘˜2<๐‘—2, being left and right singular vectors of ๐ด, respectively. Then as in (3.12) ๎€ทยฑ๐ฎ๐‘˜1,ยฑ๐ฎ๐‘˜2,โ€ฆ,๐ฎ1,๐ฎ๐‘˜1+1,โ€ฆ,๐ฎ๐‘˜2โˆ’1,๐ฎ2,๐ฎ๐‘˜2+1,โ€ฆ,๐ฎ๐‘š๎€ธ๎‚ฮฃโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐ฏโˆ—๐‘—1๐ฏโˆ—๐‘—2โ‹ฎ๐ฏโˆ—1๐ฏโˆ—๐‘—1+1โ‹ฎ๐ฏโˆ—๐‘—2โˆ’1๐ฏโˆ—2๐ฏโˆ—๐‘—2+1โ‹ฎ๐ฏโˆ—๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(3.49) with ๎‚ฮฃ=diag(|๐›ผ1|,|๐›ผ2|,0,โ€ฆ,0). It is clear that the matrix in (3.49) can be written as ๐‘ˆ๐‘ƒ1๐‘ƒ2๎‚ฮฃ๐‘„2๐‘„1๐‘‰โˆ—,(3.50) where ๐‘ƒ1,๐‘ƒ2,๐‘„1,๐‘„2 are permutation matrices of the form ๐‘ƒ1=๎‚ต๐‘Š๐‘˜1๐ผ๐‘šโˆ’๐‘˜1๎‚ถ,๐‘ƒ2=โŽ›โŽœโŽœโŽ1๐‘Š๐‘˜2๐ผ๐‘šโˆ’(๐‘˜2+1)โŽžโŽŸโŽŸโŽ ,๐‘„1=๎‚ต๐‘Š๐‘—1๐ผ๐‘›โˆ’๐‘—1๎‚ถ,๐‘„2=โŽ›โŽœโŽœโŽ1๐‘Š๐‘—2๐ผ๐‘›โˆ’(๐‘—2+1)โŽžโŽŸโŽŸโŽ .(3.51) From (3.50), the matrix in (3.49) is โŽ›โŽœโŽœโŽœโŽœโŽ๎€ท๐›ผ1๎€ธ๐‘˜1,๐‘—1๎€ท๐›ผ2๎€ธ๐‘˜2,๐‘—2โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,(3.52) where if ๐›ผ1,๐›ผ2 are negative, we multiply ๐ฎ๐‘˜1,๐ฎ๐‘˜2 by minus one. Thus, ๐ต=๐ด+๐›ผ1๐ฎ๐‘˜1๐ฏโˆ—๐‘—1+๐›ผ2๐ฎ๐‘˜2๐ฏโˆ—๐‘—2๎‚€ฮ›0๎‚๐‘‰=๐‘ˆโˆ—,(3.53) where โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œŽฮ›=1โ‹ฑ๐œŽ๐‘˜1โ‹ฏ๐›ผ๐‘˜1,๐‘—1๐œŽโ‹ฑโ‹ฎ๐‘—1โ‹ฑ๐œŽ๐‘˜2โ‹ฏ๐›ผ๐‘˜2,๐‘—2๐œŽโ‹ฑโ‹ฎ๐‘—2โ‹ฑ๐œŽ๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(3.54) By permuting rows and columns of ฮ› it follows that singular values of (ฮ›0) are ๐œŽ1,โ€ฆ,๐œŽ๐‘˜1โˆ’1,๐œŽ๐‘˜1+1,โ€ฆ,๐œŽ๐‘—1โˆ’1,๐œŽ๐‘—1+1,โ€ฆ,๐œŽ๐‘˜2โˆ’1,๐œŽ๐‘˜2+1,โ€ฆ,๐œŽ๐‘—2โˆ’1,๐œŽ๐‘—2+1,โ€ฆ,๐œŽ๐‘›(3.55) together with the singular values of ๐ถ1=๎‚ต๐œŽ๐‘˜1๐›ผ๐‘˜1,๐‘—10๐œŽ๐‘—1๎‚ถ,๐ถ2=๎‚ต๐œŽ๐‘˜2๐›ผ๐‘˜2,๐‘—20๐œŽ๐‘—2๎‚ถ.(3.56) Is immediate that the singular values of these matrices are ๎‚๐œŽ๐‘˜1=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ21+๐œŽ2๐‘˜1+๐œŽ2๐‘—1+๎‚™๎‚€๐›ผ21+๐œŽ2๐‘˜1+๐œŽ2๐‘—1๎‚2โˆ’4๐œŽ2๐‘˜1๐œŽ2๐‘—12โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2,๎‚๐œŽ๐‘—1=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ21+๐œŽ2๐‘˜1+๐œŽ2๐‘—1โˆ’๎‚™๎‚€๐›ผ21+๐œŽ2๐‘˜1+๐œŽ2๐‘—1๎‚2โˆ’4๐œŽ2๐‘˜1๐œŽ2๐‘—12โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2,๎‚๐œŽ๐‘˜2=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ22+๐œŽ2๐‘˜2+๐œŽ2๐‘—2+๎‚™๎‚€๐›ผ22+๐œŽ2๐‘˜2+๐œŽ2๐‘—2๎‚2โˆ’4๐œŽ2๐‘˜2๐œŽ2๐‘—22โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2,๎‚๐œŽ๐‘—2=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ22+๐œŽ2๐‘˜2+๐œŽ2๐‘—2โˆ’๎‚™๎‚€๐›ผ22+๐œŽ2๐‘˜2+๐œŽ2๐‘—2๎‚2โˆ’4๐œŽ2๐‘˜2๐œŽ2๐‘—22โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2.(3.57) The matrix ๐ต=๐ด+๐›ผ1๐ฎ๐‘˜1๐ฏโˆ—๐‘—1+๐›ผ2๐ฎ๐‘˜2๐ฏโˆ—๐‘—2 can be written as ๎‚๐‘ˆ๎‚ต๎‚ฮ›0๎‚ถ๎‚๐‘‰๐ต=โˆ—๎‚with๐‘ˆ=๐‘ˆ1๐‘ˆ2,๎‚๐‘‰=๐‘‰1๐‘‰2,๎‚โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œŽฮ›=1โ‹ฑ๎‚๐œŽ๐‘˜1๎‚๐œŽ๐‘—1โ‹ฑ๎‚๐œŽ๐‘˜2๎‚๐œŽ๐‘—2โ‹ฑ๐œŽ๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(3.58) where the ๎‚๐œŽโ€™s are not ordered, ๐‘ˆ1=๎€ท๐ฎ1,๐ฎ2โ‹ฏ๐ฎ๐‘˜1,๐ฎ๐‘—1โ‹ฏ๐ฎ๐‘˜1+1,๐ฎ๐‘—1+1โ‹ฏ๐ฎ๐‘˜2,๐ฎ๐‘—2โ‹ฏ๐ฎ๐‘˜2+1,๐ฎ๐‘—2+1โ‹ฏ๐ฎ๐‘š๎€ธ,๐‘‰1=๎€ท๐ฏ1,๐ฏ2โ‹ฏ๐ฏ๐‘˜1,๐ฏ๐‘—1โ‹ฏ๐ฏ๐‘˜1+1,๐ฏ๐‘—1+1โ‹ฏ๐ฏ๐‘˜2,๐ฏ๐‘—2โ‹ฏ๐ฏ๐‘˜2+1,๐ฏ๐‘—2+1โ‹ฏ๐ฏ๐‘›๎€ธ,๐‘ˆ2=๎€ท๐ž1โ‹ฏ๐ž๐‘˜1โˆ’1,ฬƒ๐ฎ๐‘˜1ฬƒ๐ฎ๐‘˜1+1,๐ž๐‘˜1+2โ‹ฏ๐ž๐‘˜2โˆ’1,ฬƒ๐ฎ๐‘˜2,ฬƒ๐ฎ๐‘˜2+1,๐ž๐‘˜2+2โ‹ฏ๐ž๐‘š๎€ธ,๐‘‰2=๎€ท๐ž1โ‹ฏ๐ž๐‘˜1โˆ’1,ฬƒ๐ฏ๐‘˜1ฬƒ๐ฏ๐‘˜1+1,๐ž๐‘˜1+2โ‹ฏ๐ž๐‘˜2โˆ’1,ฬƒ๐ฏ๐‘˜2,ฬƒ๐ฏ๐‘˜2+1,๐ž๐‘˜2+2โ‹ฏ๐ž๐‘›๎€ธ.(3.59) We have proved the following result.

Theorem 3.5. Let A be an ๐‘šร—๐‘› matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿโ‰ฅ0,๐‘Ÿ=min{๐‘š,๐‘›} and SVD ๐ด=๐‘ˆฮฃ๐‘‰โˆ—, where ๎€ท๐ฎ๐‘ˆ=1,๐ฎ2โ‹ฏ๐ฎ๐‘˜1โ‹ฏ๐ฎ๐‘˜2โ‹ฏ๐ฎ๐‘š๎€ธ๎€ท๐ฏ,๐‘‰=1,๐ฏ2โ‹ฏ๐ฏ๐‘—1โ‹ฏ๐ฏ๐‘—2โ‹ฏ๐ฏ๐‘›๎€ธ,(3.60)๐‘˜๐‘–,๐‘—๐‘–โ‰ค๐‘Ÿ,๐‘˜๐‘–โ‰ ๐‘—๐‘–, are ๐‘šร—๐‘š and ๐‘›ร—๐‘› unitary matrices, respectively, and ฮฃ=diag{๐œŽ1,๐œŽ2,โ€ฆ,๐œŽ๐‘Ÿ}. Let ๐›ผ1,๐›ผ2 be real numbers. Then ๐ต=๐ด+๐›ผ1๐ฎ๐‘˜1๐ฏโˆ—๐‘—1+๐›ผ2๐ฎ๐‘˜2๐ฏโˆ—๐‘—2 has singular values ๐‘Ÿ๎š๐‘ž=1๐‘žโ‰ ๐‘˜๐‘–,๐‘—๐‘–๎€ฝ๐œŽ๐‘ž๎€พโˆช๎€ฝ๎‚๐œŽ๐‘˜1,๎‚๐œŽ๐‘—1,๎‚๐œŽ๐‘˜2,๎‚๐œŽ๐‘—2๎€พ,(3.61)๎‚๐œŽ๐‘˜๐‘–=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ2๐‘–+๐œŽ2๐‘˜๐‘–+๐œŽ2๐‘—๐‘–+๎‚™๎‚€๐›ผ2๐‘–+๐œŽ2๐‘˜๐‘–+๐œŽ2๐‘—๐‘–๎‚2โˆ’4๐œŽ2๐‘˜i๐œŽ2๐‘—๐‘–2โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2,(3.62)๎‚๐œŽ๐‘—๐‘–=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ2๐‘–+๐œŽ2๐‘˜๐‘–+๐œŽ2๐‘—๐‘–โˆ’๎‚™๎‚€๐›ผ2๐‘–+๐œŽ2๐‘˜๐‘–+๐œŽ2๐‘—๐‘–๎‚2โˆ’4๐œŽ2๐‘˜๐‘–๐œŽ2๐‘—๐‘–2โŽžโŽŸโŽŸโŽŸโŽŸโŽ 1/2(3.63)๐‘–=1,2. The singular vectors are given by ฬƒ๐ฎ๐‘ž=๐ฎ๐‘ž,๐‘ž=1,โ€ฆ,๐‘š,๐‘žโ‰ ๐‘˜๐‘–,๐‘žโ‰ ๐‘—๐‘–ฬƒ๐ฎ,๐‘–=1,2,๐‘˜1=๐œ”(1)11๐ฎ๐‘˜1+๐œ”(1)21๐ฎ๐‘—1,ฬƒ๐ฎ๐‘—1=๐œ”(1)12๐ฎ๐‘˜1+๐œ”(1)22๐ฎ๐‘—1,ฬƒ๐ฎ๐‘˜2=๐œ”(2)11๐ฎ๐‘˜2+๐œ”(2)21๐ฎ๐‘—2,ฬƒ๐ฎ๐‘—2=๐œ”(2)12๐ฎ๐‘˜2+๐œ”(2)22๐ฎ๐‘—2,(3.64) where the coefficients ๐œ”โ€™s are obtained as before, and ฬƒ๐ฏ๐‘ž=๐ฏ๐‘ž,๐‘ž=1,โ€ฆ,๐‘›,๐‘žโ‰ ๐‘˜๐‘–,๐‘žโ‰ ๐‘—๐‘–ฬƒ๐ฏ,๐‘–=1,2,๐‘˜1=๐œ(1)11๐ฏ๐‘˜1+๐œ(1)21๐ฏ๐‘—1,ฬƒ๐ฏ๐‘—1=๐œ(1)12๐ฏ๐‘˜1+๐œ(1)22๐ฏ๐‘—1,ฬƒ๐ฏ๐‘˜2=๐œ(2)11๐ฏ๐‘˜2+๐œ(2)21๐ฏ๐‘—2,ฬƒ๐ฏ๐‘—2=๐œ(2)12๐ฏ๐‘˜2+๐œ(2)22๐ฏ๐‘—2.(3.65)

Now, as in Lemma 2.1 in Section 2, we look for a condition to preserve nonnegativity when we deal with mixed perturbations. Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}, and with singular value decomposition ๐ด=๐‘ˆฮฃ๐‘‰โˆ—, where ๎€ท๐ฎ๐‘ˆ=1โˆฃโ‹ฏโˆฃ๐ฎ๐‘โˆฃโ‹ฏโˆฃ๐ฎ๐‘š๎€ธ๎€ท๐ฏ,๐‘‰=1โˆฃโ‹ฏโˆฃ๐ฏ๐‘žโˆฃโ‹ฏโˆฃ๐ฏ๐‘›๎€ธ๎€ฝ๐œŽ,๐‘,๐‘žโ‰ค๐‘Ÿ,ฮฃ=diag1,๐œŽ2,โ€ฆ,๐œŽ๐‘Ÿ๎€พ.(3.66) Then ๐ด+๐›ผ๐ฎ๐‘๐ฏ๐‘‡๐‘ž is nonnegative for ๐›ผ>0 if ๐‘Ž๐‘–๐‘˜+๐›ผ(๐ฎ๐‘๐ฏ๐‘‡๐‘ž)๐‘–๐‘˜โ‰ฅ0,๐‘–=1,โ€ฆ,๐‘š,๐‘˜=1,โ€ฆ,๐‘›. If (๐ฎ๐‘๐ฏ๐‘‡๐‘ž)๐‘–๐‘˜<0, then ๐‘Ž๐‘–๐‘˜๎€ท๐ฎ+๐›ผ๐‘๐ฏ๐‘‡๐‘ž๎€ธ๐‘–๐‘˜โ‰ฅ0i๏ฌ€0<๐›ผโ‰คmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘๐ฏ๐‘‡๐‘ž๎€ธ๐‘–๐‘˜||.(3.67) Then we have the following result.

Lemma 3.6. Let ๐ด be an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}. Let ๐›ผ be in the interval ๎ƒฉ0,min๐‘,๐‘žmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘๐ฏ๐‘‡๐‘ž๎€ธ๐‘–๐‘˜||๎ƒญ.(3.68) Then ๐ด+๐›ผ๐ฎ๐‘๐ฏ๐‘‡๐‘ž,1โ‰ค๐‘โ‰ค๐‘š,1โ‰ค๐‘žโ‰ค๐‘›,๐‘โ‰ ๐‘ž, is nonnegative with singular values ๐œŽ1,โ€ฆ,๐œŽ๐‘โˆ’1,๎‚๐œŽ๐‘,๐œŽ๐‘+1,โ€ฆ,๐œŽ๐‘žโˆ’1,๎‚๐œŽ๐‘ž,๐œŽ๐‘ž+1,โ€ฆ,๐œŽ๐‘Ÿ,(3.69) where ๎‚๐œŽ๐‘ and ๎‚๐œŽ๐‘ž are defined as in (3.62) and (3.63), respectively.

Example 3.7. Let ๐ด=(123456) be with singular values ๐œŽ1=9.5255,๐œŽ2=0.5143.๐ด=๐‘ˆฮฃ๐‘‰๐‘‡, where ๎ƒฉ๎ƒช๎‚€๎‚๐‘ˆ=0.22985โˆ’0.883460.408250.52474โˆ’0.24078โˆ’0.816500.819640.401900.40825,๐‘‰=0.619630.784890.78489โˆ’0.61963.(3.70) Then, from (3.68) we have for ๐‘=1,2,3;๐‘ž=1,2,min๐‘,๐‘žmin๐‘–,๐‘˜๐‘Ž๐‘–๐‘˜||๎€ท๐ฎ๐‘๐ฏ๐‘‡๐‘ž๎€ธ๐‘–๐‘˜||=1.8268(3.71) and ๐›ผโˆˆ(0,1.8268]. For ๐›ผ=1.8268, we have ๐ด+๐›ผ๐ฎ1๐ฏ๐‘‡2=๎ƒฉ๎ƒช๎ƒฉ๎ƒช=๎ƒฉ๎ƒช123456+1.82680.18041โˆ’0.142420.41186โˆ’0.325140.64333โˆ’0.507871.32961.73983.75243.4066.17525.0722(3.72) is nonnegative with singular values ๎‚๐œŽ1=9.6996 and