Nonnegativity Preservation under Singular Values Perturbation

Abstract

We study how singular values and singular vectors of a matrix 𝐴 change, under matrix perturbations of the form 𝐴+𝛼𝐮_𝑖𝐯^∗𝑖 and 𝐴+𝛼𝐮_𝑝𝐯^∗𝑞, 𝑝≠𝑞, where 𝛼∈ℝ, 𝐴 is an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}, and 𝐮_𝑗,𝐯_𝑘,𝑗=1,…,𝑚;𝑘=1,…,𝑛, 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 𝐴∈ℂ^𝑚×𝑛 is a factorization 𝐴=𝑈Σ𝑉^∗, where Σ=diag{𝜎₁,𝜎₂,…,𝜎_𝑟}∈ℝ^𝑚×𝑛,𝑟=min{𝑚,𝑛},𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟≥0 and both 𝑈∈ℂ^𝑚×𝑚 and 𝑉∈ℂ^𝑛×𝑛 are unitary. The diagonal entries of Σ are called the singular values of 𝐴. The columns 𝐮_𝑗 of 𝑈 are called left singular vectors of 𝐴 and the columns 𝐯_𝑗 of 𝑉 are called right singular vectors of 𝐴. Every 𝐴∈ℂ^𝑚×𝑛 has a singular value decomposition 𝐴=𝑈Σ𝑉^∗ and the following relations hold: 𝐴𝐯_𝑗=𝜎_𝑗𝐮_𝑗,𝐴^∗𝐮_𝑗=𝜎_𝑗𝐯_𝑗, and 𝐮^∗𝑗𝐴𝐯_𝑗=𝜎_𝑗. If 𝐴∈ℝ^𝑚×𝑛, then 𝑈 and 𝑉 may be taken to be real (see [1]).

Let 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛} and left and right singular vectors 𝐮_𝑗,𝐯_𝑘,𝑗=1,…,𝑚;𝑘=1,…,𝑛, respectively. In this paper we study how singular values and singular vectors of 𝐴 change, under matrix perturbations of the form 𝐴+𝛼𝐮_𝑖𝐯^∗𝑖 and 𝐴+𝛼𝐮_𝑝𝐯^∗𝑞,𝑝≠𝑞,𝛼∈ℝ. Perturbations of the form 𝐴+𝛼𝐮_𝑖𝐯^∗𝑖 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 𝑟 eigenvalues of a matrix of order 𝑛,𝑟≤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 𝑚×𝑛 matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟≥0,𝑟=min{𝑚,𝑛}. Let 𝑈=𝐮₁∣𝐮₂∣⋯∣𝐮_𝑝,𝑉=𝐯₁∣𝐯₂∣⋯∣𝐯_𝑝,𝑝≤𝑟,(1.1) be matrices of order 𝑚×𝑝 and 𝑛×𝑝, whose columns are the left and right singular vectors, respectively, corresponding to 𝜎_𝑖,𝑖=1,…,𝑝. Let 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑝} with 𝜎_𝑖+𝑑_𝑖≥0. Then 𝐴+UDV^∗ has singular values 𝜎₁+𝑑₁,…,𝜎_𝑝+𝑑_𝑝,𝜎_𝑝+1,…,𝜎_𝑟.(1.2)

Note that the singular values of 𝐴+UDV^∗ are not necessarily in nondecreasing order. However we can reorderer them by using an appropriate permutation.

Corollary 1.2. Let 𝐴 be an 𝑚×𝑛 matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟≥0,𝑟=min{𝑚,𝑛}. Let 𝐮_𝑖 and 𝐯_𝑖, respectively, the left and right singular vectors corresponding to 𝜎_𝑖,𝑖=1,…,𝑟. Let 𝛼∈ℝ such that 𝛼+𝜎_𝑖≥0,𝑖=1,…,𝑟. Then 𝐴+𝛼𝐮_𝑖𝐯^∗𝑖 has singular values 𝜎₁,…,𝜎_𝑖−1,𝜎_𝑖+𝛼,𝜎_𝑖+1,…,𝜎_𝑟.(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 𝜅₂(𝐴)=𝜎₁/𝜎_𝑟, then we may choose 0<𝛼₁≤𝜎₁−𝜎₂ and 0<𝛼_𝑟≤𝜎_𝑟−1−𝜎_𝑟 in such a way that 𝜅₂𝐴−𝛼₁𝐮₁𝐯^∗1+𝛼_𝑟𝐮_𝑟𝐯^∗𝑟=𝜎¹−𝛼₁𝜎_𝑟+𝛼_𝑟<𝜅₂(𝐴).(1.4)

The paper is organized as follows. In Section 2 we consider perturbations of the form 𝐴+𝛼𝐮_𝑖𝐯^∗𝑖, 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 𝐴+𝛼𝐮_𝑝𝐯^∗𝑞,𝑝≠𝑞, 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 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}. In this section we consider perturbations 𝐴+𝛼𝐮_𝑖𝐯^𝑇𝑖, which preserve nonnegativity. Note that if 𝐴 is an 𝑚×𝑛 nonnegative matrix, then the left and right singular vectors 𝐮₁ and 𝐯₁, corresponding to the maximal singular value 𝜎₁,respectively, are nonnegative. Hence, in this case, the matrix 𝐴+𝛼𝐮₁𝐯^𝑇1 is nonnegative for all 𝛼>0.

Now, let us consider the perturbation 𝐴+𝛼𝐮_𝑖𝐯^𝑇𝑖, with 𝑖>1. Let 𝐮_𝑠 and 𝐯_𝑠 be the left and right singular vectors corresponding to 𝜎_𝑠,𝑠>1, respectively. Let 𝛼>0 and let the entry in position (𝑖,𝑘) of 𝐮_𝑠𝐯^𝑇𝑠 be negative. That is, (𝐮_𝑠𝐯^𝑇𝑠)_𝑖𝑘<0. Then if 𝐴=(𝑎_𝑖𝑘) is nonnegative, 𝑎_𝑖𝑘+𝛼𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘≥0iff0<𝛼≤𝑎^𝑖𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||.(2.1) Thus, to preserve the nonnegativity of 𝐴 it is enough to choose 𝛼 in the interval 0,min^𝑖,𝑘𝑎_𝑖𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||,(2.2) provided that 𝑎_𝑖𝑘>0, otherwise (𝐮_𝑠𝐯^𝑇𝑠)_𝑖𝑘 must be zero. Then from (2.2) and Theorem 1.1 we have the following result.

Lemma 2.1. Let 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}. Let 𝛼 be in the interval 0,min^𝑠min_𝑖,𝑘𝑎_𝑖𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||.(2.3) Then 𝐴+𝛼𝐮_𝑠𝐯^𝑇𝑠,𝑠=2,…,𝑟, is nonnegative with singular values 𝜎₁,…,𝜎_𝑠−1,𝜎_𝑠+𝛼,𝜎_𝑠+1,…,𝜎_𝑟.(2.4)

Remark 2.2. It is clear that if in Lemma 2.1, 𝛼 is taken in 0,min^𝑠min_𝑖,𝑘𝑎_𝑖𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||,(2.5) then 𝐴+𝛼𝐮_𝑠𝐯^𝑇𝑠,2≤𝑠≤𝑟, is positive with singular values 𝜎₁,…,𝜎_𝑠−1,𝜎_𝑠+𝛼,𝜎_𝑠+1,…,𝜎_𝑟.(2.6) Moreover, for 𝛼 in intervals in Lemma 2.1 and this remark, the nonnegativity is obtained independently of the chosen singular vectors 𝐮_𝑠,𝐯_𝑠.

A more handle interval for 𝛼 is given by the following lemma.

Lemma 2.3. Let 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}. Let 𝛼 be in the interval 0,𝜎^𝑟min_𝑖,𝑘𝑎_𝑖𝑘𝜎₁+∑^{𝑟−1𝑗=1}𝜎_𝑗.(2.7) Then 𝐴+𝛼𝐮_𝑠𝐯^𝑇𝑠,2≤𝑠≤𝑟, is nonnegative with singular values 𝜎₁,…,𝜎_𝑠−1,𝜎_𝑠+𝛼,𝜎_𝑠+1,…,𝜎_𝑟.(2.8)

Proof. From (2.2) and since |𝑎_𝑖𝑘|≤𝜎₁, see [1, Corollary  3.1.3], we have max_𝑖,𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||=max_𝑖,𝑘|||||𝑎_𝑖𝑘𝜎_𝑠−^𝑟_𝑗=_{1,𝑗≠𝑠}𝜎_𝑗𝜎_𝑠𝐮_𝑗𝐯^𝑇𝑗_𝑖𝑘≤1|||||𝜎_𝑠max_𝑖,𝑘||𝑎_𝑖𝑘||+^𝑟_𝑗=_{1,𝑗≠𝑠}max_𝑖,𝑘𝜎_𝑗𝜎_𝑠≤1||𝐮_𝑗𝐯^𝑇𝑗_𝑖𝑘||𝜎_𝑠max_𝑖,𝑘||𝑎_𝑖𝑘||+^𝑟_𝑗=_{1,𝑗≠𝑠}max_𝑖,𝑘𝜎_𝑗𝜎_𝑠≤𝜎¹𝜎_𝑠+^𝑟_𝑗=_{1,𝑗≠𝑠}𝜎_𝑗𝜎_𝑠≤1𝜎_𝑠2𝜎¹+^𝑟_𝑗=_{2,𝑗≠𝑠}𝜎_𝑗=1𝜎_𝑠2𝜎¹−𝜎_𝑠+^𝑟_𝑗=2𝜎_𝑗=2𝜎₁𝜎_𝑠−1+^𝑟_𝑗=2𝜎_𝑗𝜎_𝑠≤2𝜎¹𝜎_𝑟−1+^𝑟_𝑗=2𝜎_𝑗𝜎_𝑟=𝜎¹𝜎_𝑟+^𝑟−1_𝑗=1𝜎_𝑗𝜎_𝑟.(2.9) Then 1max_𝑖,𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||≥𝜎₁+∑𝜎^{𝑟𝑟−1𝑗=1}𝜎_𝑗min_𝑖,𝑘,𝑎_𝑖𝑘max_𝑖,𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||≥𝜎^𝑟min_𝑖,𝑘𝑎_𝑖𝑘𝜎₁+∑^{𝑟−1𝑗=1}𝜎_𝑗.(2.10) Hence we have 0,𝜎^𝑟min_𝑖,𝑘𝑎_𝑖𝑘𝜎₁+∑^{𝑟−1𝑗=1}𝜎_𝑗⊆0,min^𝑖,𝑘𝑎_𝑖𝑘max_𝑖,𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||⊆0,min^𝑖,𝑘𝑎_𝑖𝑘0,𝜎_𝑟||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||,min_𝑖,𝑘𝑎_𝑖𝑘𝜎₁+∑^{𝑟−1𝑗=1}𝜎_𝑗⊆0,min^𝑠min_𝑖,𝑘𝑎_𝑖𝑘||𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘||.(2.11) Then, from Lemma 2.1 the result follows.

Remark 2.4. For positive 𝐴=(𝑎_𝑖𝑗), and 𝛼∈ℝ, we repeat the arguments from Lemmas 2.1 and 2.3 to obtain that if 𝛼 is in the interval −𝜎^𝑟min_𝑖,𝑘𝑎_𝑖𝑘𝜎₁+∑^{𝑟−1𝑗=1}𝜎_𝑗,𝜎^𝑟min_𝑖,𝑘𝑎_𝑖𝑘𝜎₁+∑^{𝑟−1𝑗=1}𝜎_𝑗,(2.12) then 𝐴+𝛼𝐮_𝑠𝐯^𝑇𝑠,𝑠=2,…,𝑟, is nonnegative with singular values 𝜎₁,…,𝜎_𝑠−1,||𝜎_𝑠+𝛼||,𝜎_𝑠+1,…,𝜎_𝑟.(2.13)

Now we consider rank-2 perturbations, 𝐴+UDV^∗, where 𝑈=(𝐮_𝑠,𝐮_𝑡),𝐷=diag{𝛼₁,𝛼₂},𝑉=(𝐯_𝑠,𝐯_𝑡). That is, perturbations of the form 𝐴+𝛼₁𝐮_𝑠𝐯^𝑇𝑠+𝛼₂𝐮_𝑡𝐯^𝑇𝑡 as in Theorem 1.1. Let 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}. Then 𝐴+𝛼₁𝐮_𝑠𝐯^𝑇𝑠+𝛼₂𝐮_𝑡𝐯^𝑇𝑡 will be nonnegative if 𝑎_𝑖𝑘+𝛼₁𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘+𝛼₂𝐮_𝑡𝐯^𝑇𝑡_𝑖𝑘≥0,𝑖=1,…,𝑚;𝑗=1,…,𝑛.(2.14) From the family of straight lines 𝛼₂=−𝐮^𝑠𝐯^𝑇𝑠_𝑖𝑘𝐮_𝑡𝐯^𝑇𝑡_𝑖𝑘𝛼₁−𝑎^𝑖𝑘𝐮_𝑡𝐯^𝑇𝑡_𝑖𝑘,𝑖=1,…,𝑚;𝑗=1,…,𝑛,(2.15) it follows that they intersect the axes 𝛼₁ and 𝛼₂ at points −𝐮_𝑠𝑎𝐯^{𝑖𝑘𝑇𝑠}_𝑖𝑘,0,0,𝑎^𝑖𝑘𝐮_𝑠𝐯^𝑇𝑠_𝑖𝑘,(2.16) where (𝐮_𝑠𝐯^𝑇𝑠)_𝑖𝑘≠0 and (𝐮_𝑡𝐯^𝑇𝑡)_𝑖𝑘≠0, respectively. Let 𝐸=max_{𝐮𝑖,𝑘𝑠𝐯𝑇𝑠𝑖𝑘>0}𝐹=max_{𝐮𝑖,𝑘𝑡𝐯𝑇𝑡𝑖𝑘>0}−𝐮_𝑠𝑎𝐯^{𝑖𝑘𝑇𝑠}_𝑖𝑘,𝐺=min_{𝐮𝑖,𝑘𝑠𝐯𝑇𝑠𝑖𝑘<0}−𝐮_𝑡𝑎𝐯^{𝑖𝑘𝑇𝑡}_𝑖𝑘,𝐻=min_{𝐮𝑖,𝑘𝑡𝐯𝑇𝑡𝑖𝑘<0}−𝐮_𝑠𝑎𝐯^{𝑖𝑘𝑇𝑠}_𝑖𝑘,−𝐮_𝑡𝑎𝐯^{𝑖𝑘𝑇𝑡}_𝑖𝑘.(2.17) Let 𝑅₁=𝛼₁,𝛼₂∶𝐸≤𝛼₁≤0∧−𝐹𝐸𝛼¹+𝐹≤𝛼₂≤−𝐻𝐸𝛼¹𝑅+𝐻₂,=𝛼₁,𝛼₂∶0≤𝛼₁≤𝐺∧−𝐹𝐺𝛼¹+𝐹≤𝛼₂≤−𝐻𝐺𝛼¹+𝐻.(2.18) Then 𝐴+𝛼₁𝐮_𝑠𝐯^𝑇𝑠+𝛼₂𝐮_𝑡𝐯^𝑇𝑡 is nonnegative for (𝛼₁,𝛼₂)∈𝑅₁∪𝑅₂.

A more handle region for (𝛼₁,𝛼₂) is given by the following lemma.

Lemma 2.5. Let 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}. Let 𝑃₁=(𝐸,0),𝑃₂=(0,𝐹),𝑃₃=(𝐺,0),𝑃₄=(0,𝐻) be the intersection points in (2.17). Let 𝛼₁,𝛼₂∈𝑆=(𝑥,𝑦)∶‖(𝑥,𝑦)‖₁≤min_𝑘=1,2,3,4‖‖𝑃_𝑘‖‖₁,(2.19) where ‖⋅‖₁ is the 𝑙₁ norm. Then 𝐴+𝛼₁𝐮_𝑠𝐯^𝑇𝑠+𝛼₂𝐮_𝑡𝐯^𝑇𝑡,2≤𝑠,𝑡≤𝑟, is nonnegative with singular values 𝜎₁,…,|𝜎_𝑠+𝛼₁|,…,|𝜎_𝑡+𝛼₂|,…,𝜎_𝑟.

Example 2.6. Let 𝐴=154269484,(2.20) with singular values 𝜎₁=15.5687298,𝜎₂=3.9581084, and 𝜎₃=0.9736668. Let 𝑈=𝐮₂∣𝐮₃,𝑉=𝐯₂∣𝐯₃,𝐷=diag𝛼₁,𝛼₂,(2.21) where 𝐮𝐯₂=⎛⎜⎜⎝−000..45.7223168940868214387⎞⎟⎟⎠,𝐮₃=⎛⎜⎜⎝−000..81.465756580370115984₂2⎞⎟⎟⎠,=⎛⎜⎜⎝−000..07.6246546769519792951⎞⎟⎟⎠,𝐯₃=⎛⎜⎜⎝−000..22.9991120566568414160⎞⎟⎟⎠.(2.22) From (2.17) we compute 𝐸,𝐹,𝐺,𝐻. Then, the intersection points are (−12.7300876,0),(7.1058357,0),(0,−7.9224079),(0,1.2681734),(2.23) and 𝑆={(𝑥,𝑦)∶‖(𝑥,𝑦)‖₁≤1.2681734}. Thus, from Lemma 2.5, for (𝛼₁,𝛼₂)=(−1/2,1/2) we have 𝐴+𝛼₁𝐮₂𝐯^𝑇2+𝛼₂𝐮₃𝐯^𝑇3=053...6290030983750268...217607766036374...972639947646,(2.24) with singular values 𝜎₁,𝜎₂+𝛼₁,𝜎₃+𝛼₂.

Remark 2.7. Let 𝐴 be an 𝑚×𝑛 complex matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛} and singular value decomposition 𝐴=𝑈Σ𝑉^∗. In [11] it was defined the concept of energy of 𝐴 as ℰ(𝐴)=𝜎₁+𝜎₂+⋯+𝜎_𝑟. If 𝐴 is positive, then as an application of the rank-2 perturbation result, from Lemma 2.5 and Remark 2.4, we may construct, for 0<𝛼≤min𝜎^𝑟min_𝑖,𝑘𝑎_𝑖𝑘𝜎₁+∑^{𝑟−1𝑗=1}𝜎_𝑗,𝜎_𝑗,(2.25) a family of nonnegative matrices 𝐵=𝐴+𝛼𝐮_𝑖𝐯^𝑇𝑖−𝛼𝐮_𝑗𝐯^𝑇𝑗, with ℰ(𝐵)=ℰ(𝐴). Now, suppose 𝐴 is nonnegative with ℰ(𝐴)≤𝐶, where 𝐶 is an upper bound. Then from (1.2), by taking 𝛼=𝐶−ℰ(𝐴) we may construct a family of nonnegative matrices 𝐵=𝐴+𝛼𝐮₁𝐯^𝑇1 with ℰ(𝐵)=ℰ(𝐴)+𝛼=𝐶.

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 𝑛×𝑛 matrix 𝐴=(𝑎_𝑖𝑗) is said to be with constant row sums if ∑^𝑛𝑗=1𝑎_𝑖𝑗=𝛼,𝑖=1,2,…,𝑛. We denote by CS_𝛼 the set of all matrices with constant row sums equal to 𝛼.

Lemma 2.8. Let 𝐴 be an 𝑛×𝑛 irreducible doubly stochastic matrix and let 𝐴𝐱=𝜆𝐱,𝐱^𝑇=(𝑥₁,…,𝑥_𝑛), with 𝜆≠1. Then 𝑆(𝐱)=𝑥₁+𝑥₂+⋯+𝑥_𝑛=0.(2.26)

Proof. Since 𝐴 is doubly stochastic, then 𝑆(𝐴𝐱)=Σ𝑎_1𝑗𝑥_𝑗+Σ𝑎_2𝑗𝑥_𝑗+⋯+Σ𝑎_𝑛𝑗𝑥_𝑗=𝑥₁Σ𝑎_𝑖1+𝑥₂Σ𝑎_𝑖2+⋯+𝑥_𝑛Σ𝑎_𝑖𝑛=𝑆(𝑥),(2.27) and 𝑆(𝐴𝐱)=𝑆(𝜆𝐱)=𝜆𝑆(𝐱). Then, 𝑆(𝐱)=𝜆𝑆(𝐱),(2.28) and since 𝜆≠1,𝑆(𝐱)=0.

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

Proposition 2.9. Let 𝐴 be an 𝑛×𝑛 irreducible doubly stochastic matrix. Then, ((iii))𝐴𝐴++𝛼𝛼_1𝑖𝐮₁𝐯^𝑇1∈CS_1+𝛼1,𝐴+𝛼₁𝐮₁𝐯^𝑇1^𝑇∈CS_1+𝛼1,(iii)𝐴+𝐮_𝑖𝑛𝐯^𝑇𝑖∈CS₁,𝐴+𝛼_𝑖𝐮_𝑖𝐯^𝑇𝑖^𝑇∈CS₁;𝑖=2,…,𝑛,_𝑖=1𝛼_𝑖𝐮_𝑖𝐯^𝑇𝑖∈CS^1+𝛼1.(2.29)

Proof. Since 𝐴,𝐴^𝑇∈CS₁, then 𝐴𝐴^𝑇∈CS₁ and 𝐴^𝑇𝐴∈CS₁. Hence, the singular vectors 𝐮₁ and 𝐯₁ are 𝐮₁=𝐯₁=1√𝑛𝐞=√1𝑛(1,1,…,1)^𝑇.(2.30) Then 𝛼₁𝐮₁𝐯^𝑇1=𝛼₁𝐯₁𝐮^𝑇1=(1/𝑛)𝛼₁𝐞𝐞^𝑇 and 𝛼₁𝐮₁𝐯^𝑇1∈CS_𝛼1. Thus (i) holds. From Lemma 2.8, 𝛼_𝑖𝐮_𝑖𝐯^𝑇𝑖𝐞=𝛼_𝑖(𝐯^𝑇𝑖𝐞)𝐮_𝑖=0 and (ii) holds. From (i) and (ii) we have (iii).

Example 2.10. Consider the matrix 𝐴=⎛⎜⎜⎝4123341223411234⎞⎟⎟⎠,(2.31) which is nonnegative generalized doubly stochastic, that is, 𝐴 is nonnegative with 𝐴,𝐴^𝑇∈CS₁₀, and 𝐴 has singular values 10,2.8284,2,8284,2. Let 𝑈Σ𝑉^∗ be the singular value decomposition of 𝐴. Let 𝐷=diag{4,3,2,1}. Then 𝐵=𝐴+𝑈𝐷𝑉^∗=⎛⎜⎜⎝6124....20245841920762484602....80645059080101192461....24204158079248620246....6480595001081901⎞⎟⎟⎠(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 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗, with 𝑘≠𝑗,̇ 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 𝐴 be a 4×3 matrix with singular values 𝜎₁,𝜎₂,𝜎₃. Let 𝐴=𝑈Σ𝑉^∗ with 𝑈=(𝐮₁∣𝐮₂∣𝐮₃∣𝐮₄) and𝑉=(𝐯₁∣𝐯₂∣𝐯₃). Let 𝛼≥0. That is, 𝐴=𝐮₁∣𝐮₂∣𝐮₃∣𝐮₄⎛⎜⎜⎝𝜎¹0𝜎0₂000𝜎0₃000⎞⎟⎟⎠𝐯^∗1𝐯^∗2𝐯^∗3=³_𝑘=1𝜎_𝑘𝐮_𝑘𝐯^∗𝑘.(3.1) The matrix 𝛼𝐮₁𝐯^∗2 has the singular value decomposition: 𝛼𝐮₁𝐯^∗2=𝐮₁∣𝐮₂∣𝐮₃∣𝐮₄⎛⎜⎜⎝𝛼00000000000⎞⎟⎟⎠𝐯𝐯^∗2∗1=𝐮₁𝐯^∗3∣𝐮₂∣𝐮₃∣𝐮₄⎛⎜⎜⎝𝛼00000000000⎞⎟⎟⎠010100001𝐯𝐯^∗1∗2=𝐮₁𝐯^∗3∣𝐮₂∣𝐮₃∣𝐮₄⎛⎜⎜⎝0000𝛼0000000⎞⎟⎟⎠𝐯𝐯^∗1∗2𝐯^∗3.(3.2) Then, 𝐴+𝛼𝐮₁𝐯^∗2=𝐮₁∣𝐮₂∣𝐮₃∣𝐮₄⎛⎜⎜⎝𝜎¹0𝜎𝛼₂000𝜎0₃000⎞⎟⎟⎠𝐯^∗1𝐯^∗2𝐯^∗3.(3.3) Now we compute the singular values of the matrix 𝐶=⎛⎜⎜⎝𝜎¹0𝜎𝛼₂000𝜎0₃000⎞⎟⎟⎠,(3.4) by computing the eigenvalues of 𝐶𝐶^𝑇, where 𝐶=(^{𝜎1𝛼0𝜎}₂). Since 𝐶𝐶^𝑇=𝛼²+𝜎²¹𝛼𝜎₂𝛼𝜎₂𝜎²²,(3.5) then tr𝐶d𝐶e^𝑇t𝐶=𝐶𝛼²_𝑇+𝜎²¹+𝜎²²=𝜆₁+𝜆₂,=𝜎²¹𝜎²²=𝜆₁𝜆₂(3.6) with 𝜆₁,𝜆₂ being the eigenvalues of 𝐶𝐶^∗. Thus, we obtain 𝜆₁=𝛼²+𝜎²¹+𝜎²²+𝛼²+𝜎²¹+𝜎²²²−4𝜎²¹𝜎²²𝜆₂2,=𝛼²+𝜎²¹+𝜎²²−𝛼²+𝜎²¹+𝜎²²²−4𝜎²¹𝜎²²2.(3.7) Hence, the singular values of 𝐴+𝛼𝐮₁𝐯^∗2 are √𝜆₁,√𝜆₂,𝜎₃.

Now we generalize these results for 𝑚×𝑛 matrices.

Theorem 3.1. Let 𝐴 be an 𝑚×𝑛 matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟≥0,𝑟=min{𝑚,𝑛}, and with singular value decomposition 𝐴=𝑈Σ𝑉^∗, where 𝑈=𝐮₁∣⋯Σ∣𝐮=_𝑘d∣ia⋯g∣𝜎𝐮_𝑚1,𝑉=𝐯₁∣⋯∣𝐯_𝑗∣⋯∣𝐯_𝑛,,𝜎₂,…,𝜎_𝑟,𝑘,𝑗≤𝑟.(3.8) Let 𝛼∈ℝ. Then 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗 has singular values, 𝜎₁,…𝜎_𝑘−1,𝜎_𝑘,𝜎_𝑘+1,…,𝜎_𝑗−1,𝜎_𝑗,𝜎_𝑗+1,…,𝜎_𝑟,(3.9) where 𝜎_𝑘=⎛⎜⎜⎜⎜⎝𝛼²+𝜎^2𝑘+𝜎^2𝑗+𝛼²+𝜎^2𝑘+𝜎^2𝑗²−4𝜎^2𝑘𝜎^2𝑗2⎞⎟⎟⎟⎟⎠^1/2𝜎_𝑗,(3.10)=⎛⎜⎜⎜⎜⎝𝛼²+𝜎^2𝑘+𝜎^2𝑗−𝛼²+𝜎^2𝑘+𝜎^2𝑗²−4𝜎^2𝑘𝜎^2𝑗2⎞⎟⎟⎟⎟⎠^1/2.(3.11)

Proof. Without loss of generality we may assume that 𝑚≥𝑛 and 𝑗>𝑘. The matrix 𝛼𝐮_𝑘𝐯^∗𝑗 has a singular value decomposition 𝛼𝐮_𝑘𝐯^∗𝑗=±𝐮_𝑘∣⋯∣𝐮_𝑘−1∣𝐮₁∣𝐮_𝑘+1∣⋯∣𝐮_𝑚Σ⎛⎜⎜⎜⎜⎜⎜⎝𝐯^∗𝑗𝐯⋮_∗𝑗−1𝐯^∗1𝐯^∗𝑗+1𝐯⋮_∗𝑛⎞⎟⎟⎟⎟⎟⎟⎠,(3.12) where Σ=diag{|𝛼|,0,…,0}. The decomposition in (3.12) can be written as 𝛼𝐮_𝑘𝐯^∗𝑗=𝑈𝑃_𝑘Σ𝑄_𝑗𝑉^∗,(3.13) where 𝑃_𝑘 and 𝑄_𝑗 are 𝑚×𝑚 and 𝑛×𝑛 permutation matrices of the form 𝑃_𝑘=𝑊^𝑘0𝐼0_𝑚−𝑘,𝑄^𝑗=𝑊^𝑗0𝐼0_𝑛−𝑗,(3.14) with 𝑊_𝑙=𝐼_𝑙−211oforder𝑙.(3.15) Since Σ=𝛼⋱⋱,then𝑃_𝑘Σ𝑄_𝑗=⋱(𝛼)_𝑘,𝑗⋱,(3.16) where if 𝛼<0, we multiply the 𝐮_𝑘 vector by minus one. Then 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗=𝑈Σ+𝑃_𝑘Σ𝑄_𝑗𝑉^∗=𝑈Λ0𝑉^∗,(3.17) where Λ=⎛⎜⎜⎜⎜⎜⎝𝜎¹⋱𝜎_𝑘⋱⋯𝜎⋮𝛼_𝑗𝜎_𝑛⎞⎟⎟⎟⎟⎟⎠.(3.18) By applying row and column permutations on the matrix Σ+𝑃_𝑘Σ𝑄_𝑗, it follows from (3.17) and (3.18) that the singular values of 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗 are 𝜎₁,…,𝜎_𝑘−1,𝜎_𝑘,𝜎_𝑘+1,…,𝜎_𝑗−1,𝜎_𝑗,𝜎_𝑗+1,…,𝜎_𝑛,(3.19) where 𝜎_𝑘 and 𝜎_𝑗 are as in (3.10) and (3.11), respectively.

Observe that in Theorem 3.1, 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗 has singular values 𝜎₁,…𝜎_𝑘−1,𝜎_𝑘,𝜎_𝑘+1,…,𝜎_𝑗−1,𝜎_𝑗,𝜎_𝑗+1,…,𝜎_𝑟,(3.20) if 𝑘,𝑗≤𝑟. If 𝑟<𝑘≤𝑚, then only 𝜎_𝑗, corresponding to the right singular vector 𝐯_𝑗, changes and take the form 𝜎_𝑗=𝛼²+𝜎^2𝑗. A straightforward calculation shows that for 𝛼>0,𝜎_𝑘≤𝜎_𝑘+𝛼. Observe that 𝜎_𝑗=𝛼²+𝜎^2𝑗≤𝜎_𝑗+𝛼.(3.21)

Example 3.2. Let 𝐴=⎛⎜⎜⎝160121450310⎞⎟⎟⎠,(3.22) with singular values 𝜎₁=7.8207,𝜎₂=5.6257,𝜎₃=1.09.Let 𝐮𝐯₁₂=0.260300.703860.397760.52784^𝑇,=0.63252−0.716470.29425^𝑇,(3.23) left and right singular vectors of 𝐴, respectively. Let 𝛼=1. Then, from (3.10) and (3.11), the matrix 𝐴+𝛼𝐮₁𝐯^∗2=⎛⎜⎜⎝0161.2...1435643145356299014.3.4.86.971251317558100.30.1.172.51605157379123⎞⎟⎟⎠(3.24) has singular values 𝜎₁=7.9477,𝜎₂=5.5358 and 𝜎₃. By using Theorem 1.1 we have that 𝐴+𝐮₁𝐯^∗1+𝐮₂𝐯^∗2 has singular values 𝜎₁+1=8.8207,𝜎₂+1=6.6257,and𝜎₃=1.09.

Different from perturbations of the form 𝐴+𝛼𝐮_𝑖𝐯^𝑇𝑖, the perturbation of Theorem 3.1 affects not only the singular values 𝜎_𝑘 and 𝜎_𝑗, but also the corresponding left and right singular vectors 𝐮_𝑘 and 𝐯_𝑗. To make this modification clear, we consider again the previous discussion to Theorem 3.1: 𝐴+𝛼𝐮₁𝐯^∗2=𝐮₁∣𝐮₂∣𝐮₃∣𝐮₄⎛⎜⎜⎝𝜎¹0𝜎𝛼₂000𝜎0₃000⎞⎟⎟⎠𝐯^∗1𝐯^∗2𝐯^∗3.(3.25) Let 𝐶=𝜎0₁𝛼𝜎₂=𝜔¹¹𝜔₁₂𝜔₂₁𝜔₂₂𝜎¹0𝜎0₂𝜐¹¹𝜐₁₂𝜐₂₁𝜐₂₂^𝑇,(3.26) the SVD of 𝐶 with 𝜎₁,𝜎₂ obtained from (3.10) and (3.11), respectively. The left singular vectors of 𝐶=⎛⎜⎜⎝𝜎¹0𝜎𝛼₂000𝜎0₃000⎞⎟⎟⎠(3.27) (eigenvectors of 𝐶𝐶^𝑇) are 𝐮𝐮̃̃₁₂=𝜔₁₁𝜔₂₁00^𝑇,𝐮̃₃=𝜔₁₂𝜔₂₂00^𝑇,𝐮̃₄=0010^𝑇=𝑒₃,=0001^𝑇=𝑒₄,(3.28) and its corresponding right singular vectors (eigenvectors of 𝐶^𝑇𝐶) are 𝐯𝐯₁₂̃̃=𝜐₁₁𝜐₂₁0^𝑇,𝐯₃̃=𝜐₁₂𝜐₂₂0^𝑇,=001^𝑇=𝑒₃.(3.29) Thus, 𝐴+𝛼𝐮₁𝐯^∗2 can be written as 𝐴+𝛼𝐮₁𝐯^∗2=𝑈Σ𝑉^∗, where 𝑈==𝜔𝐮₁₁₁∣𝐮₂∣𝐮₃∣𝐮₄𝐮̃¹∣𝐮̃₂∣𝐮̃₃∣𝐮̃₄𝑉=𝐯₁𝐮₁+𝜔₂₁𝐮₂∣𝜔₁₂𝐮₁+𝜔₂₂𝐮₂∣𝐮₃∣𝐮₄,=𝜐₁₁∣𝐯₂∣𝐯₃𝐯¹̃∣𝐯₂̃∣𝐯₃̃Σ=diag𝐯¹𝜎+₁𝜐₂₁𝐯₂∣𝜐₁₂𝐯₁+𝜐₂₂𝐯₂∣𝐯₃,,𝜎₂,𝜎₃.(3.30) Now, we generalize this result. Without loss generality we assume that 𝑘<𝑗. From (3.17) and (3.18), by permuting rows and columns of the matrix Σ+𝑃_𝑘Σ𝑄_𝑗, we have 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗=𝑈₁Λ0𝑉^∗1withΛ=⎛⎜⎜⎜⎜⎜⎜⎝𝜎¹⋱𝜎_𝑘𝜎𝛼_𝑗⋱𝜎0_𝑛𝑈₁⎞⎟⎟⎟⎟⎟⎟⎠,𝑉₁=𝐮₁∣⋯∣𝐮_𝑘∣𝐮_𝑗∣⋯∣𝐮_𝑘+1∣𝐮_𝑗+1∣⋯∣𝐮_𝑚,=𝐯₁∣⋯∣𝐯_𝑘∣𝐯_𝑗∣⋯∣𝐯_𝑘+1∣𝐯_𝑗+1∣⋯∣𝐯_𝑛.(3.31) The singular vectors of (^Λ0) are obtained from the singular vectors of the 2×2 matrix (^{𝜎𝑘𝛼0𝜎}_𝑗). Let 𝜎^𝑘0𝜎𝛼_𝑗=𝜔¹¹𝜔₁₂𝜔₂₁𝜔₂₂𝜎^𝑘0𝜎0_𝑗𝜐¹¹𝜐₁₂𝜐₂₁𝜐₂₂^𝑇.(3.32) Then the unitary matrices of the singular value decomposition of (^Λ0) are 𝑈₂=⎛⎜⎜⎝𝐼𝜔¹¹𝜔₁₂𝜔₂₁𝜔₂₂𝐼⎞⎟⎟⎠=𝐞¹𝑉₂∣⋯∣𝐞_𝑘−1∣𝐮̃_𝑘∣𝐮̃_𝑘+1∣𝐞_𝑘+2∣⋯∣𝐞_𝑚,=⎛⎜⎜⎝𝐼𝑣¹¹𝑣₁₂𝑣₂₁𝑣₂₂𝐼⎞⎟⎟⎠=𝐞¹∣⋯∣𝐞_𝑘−1∣𝐯_𝑘̃∣𝐯̃_𝑘+1∣𝐞_𝑘+2∣⋯∣𝐞_𝑛,(3.33) of order 𝑚×𝑚 and 𝑛×𝑛, respectively. Then 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗=Λ𝑈=⎛⎜⎜⎜⎜⎜⎜⎝Λ0𝜎₁𝑉^∗with𝑈=𝑈₁𝑈₂,𝑉=𝑉₁𝑉₂,⋱𝜎_𝑘𝜎_𝑗⋱𝜎_𝑛⎞⎟⎟⎟⎟⎟⎟⎠,(3.34) where 𝜎_𝑘 and 𝜎_𝑗 are not ordered. Since 𝑈𝑉==𝐮𝐯¹₁∣⋯∣𝜔₁₁𝐮_𝑘+𝜔₂₁𝐮_𝑗∣𝜔₁₂𝐮_𝑘+𝜔₂₂𝐮_𝑗∣⋯∣𝐮_𝑚,∣⋯∣𝜐₁₁𝐯_𝑘+𝜐₂₁𝐯_𝑗∣𝜐₁₂𝐯_𝑘+𝜐₂₂𝐯_𝑗∣⋯∣𝐯_𝑛,(3.35) then the singular vectors corresponding to 𝜎_𝑘,𝜎_𝑗 have been modified. We have prove the following result.

Corollary 3.3. Let 𝐴 be an 𝑚×𝑛 matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛} and with singular value decomposition 𝐴=𝑈Σ𝑉^∗, where 𝑈=𝐮₁∣⋯∣𝐮_𝑘∣⋯Σ∣=𝐮d_𝑚iag,𝜎𝑉₁=𝐯₁∣⋯∣𝐯_𝑗∣⋯𝐯_𝑛,𝑘,𝑗≤𝑟,,𝜎₂,…,𝜎_𝑟.(3.36) Let 𝛼∈ℝ. Then 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗 has left singular vectors, 𝐮̃_𝑖=𝐮_𝑖,𝐮̃𝑖=1,…,𝑚,𝑖≠𝑘,𝑖≠𝑗,_𝑘𝐮̃_𝑗=𝜔₁₁𝐮_𝑘+𝜔₂₁𝐮_𝑗,=𝜔₁₂𝐮_𝑘+𝜔₂₂𝐮_𝑗(3.37) and right singular vectors 𝐯_𝑖̃=𝐯_𝑖,𝐯̃𝑖=1,…,𝑛,𝑖≠𝑘,𝑖≠𝑗,_𝑘𝐯_𝑗̃=𝜐₁₁𝐯_𝑘+𝜐₂₁𝐯_𝑗,=𝜐₁₂𝐯_𝑘+𝜐₂₂𝐯_𝑗.(3.38)

Observe that if 2×2 orthogonal matrices in (3.32) are of the same type −𝑐𝑠𝑐𝑠or𝑐𝑠−𝑠𝑐,(3.39) then a straight forward calculation shows that (3.37) and (3.38) become 𝐮̃_𝑖=𝐮_𝑖,𝐮̃𝑖=1,…,𝑚,𝑖≠𝑘,𝑖≠𝑗,_𝑘𝐮̃_𝑗=𝑐₁𝐮_𝑘−𝑠₁𝐮_𝑗,=𝑠₁𝐮_𝑘+𝑐₁𝐮_𝑗(3.40) and 𝐯_𝑖̃=𝐯_𝑖,𝐯𝑖̃=1,…,𝑛,𝑖≠𝑘,𝑖≠𝑗,_𝑘𝐯_𝑗̃=𝑐₂𝐯_𝑘−𝑠₂𝐯_𝑗,=𝑠₂𝐯_𝑘+𝑐₂𝐯_𝑗,(3.41) respectively, while if they are of different type, then (3.38) becomes 𝐯_𝑖̃=𝐯_𝑖,𝐯𝑖̃=1,…,𝑚,𝑖≠𝑘,𝑖≠𝑗,_𝑘𝐯_𝑗̃=𝑐₂𝐯_𝑘+𝑠₂𝐯_𝑗,=𝑠₂𝐯_𝑘−𝑐₂𝐯_𝑗.(3.42) From (3.10) and (3.11) it is clear that 𝜎_𝑘>𝜎_𝑗 with 𝑘<𝑗. The following result tells us how the new singular values 𝜎_𝑘,𝜎_𝑗 relate with the previous singular values 𝜎_𝑘,𝜎_𝑗.

Corollary 3.4. Let 𝐴 be an 𝑚×𝑛 matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟≥0,𝑟=min{𝑚,𝑛} and with singular value decomposition 𝐴=𝑈Σ𝑉^∗, where 𝑈=𝐮₁∣⋯∣𝐮_𝑘∣⋯Σ∣=𝐮d_𝑚iag,𝜎𝑉₁=𝐯₁∣⋯∣𝐯_𝑗∣⋯𝐯_𝑛,𝑘,𝑗≤𝑟,,𝜎₂,…,𝜎_𝑟.(3.43) Let 𝜎_𝑘,𝜎_𝑗,𝑘<𝑗, be respectively, as in (3.10) and (3.11), the singular values of 𝐴+𝛼𝐮_𝑘𝐯^∗𝑗. Then 𝜎_𝑘>𝜎_𝑘≥𝜎_𝑗>𝜎_𝑗.

Proof. Since 4𝜎^2𝑗𝛼²>0 we have 𝜎^2𝑘+𝛼²²+𝜎^4𝑗+𝜎_2𝑘2𝜎^2𝑗𝜎^2𝑘+𝛼²−4𝜎^2𝑘𝜎^2𝑗>𝜎^2𝑘+𝛼²²+𝜎^4𝑗−2𝜎^2𝑗𝜎^2𝑘+𝛼²,𝜎_2𝑘+𝜎^2𝑗+𝛼²²−4𝜎^2𝑘𝜎^2𝑗>𝜎^2𝑘+𝛼²−𝜎^2𝑗²,+𝛼²+𝜎^2𝑗−𝜎^2𝑘+𝜎^2𝑗+𝛼²²−4𝜎^2𝑘𝜎^2𝑗<2𝜎^2𝑗.(3.44) Thus 𝜎_𝑗>12𝜎^2𝑘+𝛼²+𝜎^2𝑗−𝜎^2𝑘+𝜎^2𝑗+𝛼²²−4𝜎^2𝑘𝜎^2𝑗^1/2.(3.45) In the same way, 𝜎^2𝑗+𝛼²²+𝜎^4𝑘+𝜎_2𝑘2𝜎^2𝑘𝜎^2𝑗+𝛼²−4𝜎^2𝑘𝜎^2𝑗>𝜎^2𝑗+𝛼²²+𝜎^4𝑘−2𝜎^2𝑘𝜎^2𝑗+𝛼²,𝜎_2𝑘+𝜎^2𝑗+𝛼²²−4𝜎^2𝑘𝜎^2𝑗>𝜎^2𝑘−𝜎^2𝑗+𝛼²²,+𝜎^2𝑗+𝛼²+𝜎^2𝑘+𝜎^2𝑗+𝛼²²−4𝜎^2𝑘𝜎^2𝑗>2𝜎^2𝑘.(3.46) Then 𝜎_𝑘<12𝜎^2𝑘+𝛼²+𝜎^2𝑗+𝜎^2𝑘+𝜎^2𝑗+𝛼²²−4𝜎^2𝑘𝜎^2𝑗^1/2.(3.47) Therefore, 𝜎_𝑘>𝜎_𝑘≥𝜎_𝑗>𝜎_𝑗.(3.48) Observe that 𝜎_𝑖∈(𝜎_𝑗,𝜎_𝑘),𝑖=𝑘,𝑘+1,…,𝑗. In particular for 𝑘=1 and 𝑗=𝑛, all singular values of 𝐴 are in the interval (𝜎_𝑛,𝜎₁).

Now we extend the mixed perturbation result given by Theorem 3.1 to rank-2 perturbations, that is, perturbations of the form 𝐵=𝐴+𝛼₁𝐮_𝑘1𝐯_𝑗1+𝛼₂𝐮_𝑘2𝐯_𝑗2, with 𝛼₁,𝛼₂ nonzero real numbers and 𝐮_𝑘𝑖,𝐯_𝑗𝑖,𝑖=1,2,𝑘_𝑖<𝑗_𝑖,𝑘₁<𝑗₁<𝑘₂<𝑗₂, being left and right singular vectors of 𝐴, respectively. Then as in (3.12) ±𝐮_𝑘1,±𝐮_𝑘2,…,𝐮₁,𝐮_𝑘1+1,…,𝐮_𝑘2−1,𝐮₂,𝐮_𝑘2+1,…,𝐮_𝑚Σ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐯^∗𝑗1𝐯^∗𝑗₂𝐯⋮_∗1𝐯^∗𝑗₁₊₁𝐯⋮_∗𝑗2−1𝐯^∗2𝐯^∗𝑗₂₊₁𝐯⋮_∗𝑛⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.49) with Σ=diag(|𝛼₁|,|𝛼₂|,0,…,0). It is clear that the matrix in (3.49) can be written as 𝑈𝑃₁𝑃₂Σ𝑄₂𝑄₁𝑉^∗,(3.50) where 𝑃₁,𝑃₂,𝑄₁,𝑄₂ are permutation matrices of the form 𝑃₁=𝑊^𝑘1𝐼_𝑚−𝑘1,𝑃²=⎛⎜⎜⎝1𝑊_𝑘2𝐼_{𝑚−(𝑘2+1)}𝑄₁⎞⎟⎟⎠,=𝑊^𝑗1𝐼_𝑛−𝑗1,𝑄²=⎛⎜⎜⎝1𝑊_𝑗2𝐼_{𝑛−(𝑗2+1)}⎞⎟⎟⎠.(3.51) From (3.50), the matrix in (3.49) is ⎛⎜⎜⎜⎜⎝𝛼₁_𝑘1,𝑗1𝛼₂_𝑘2,𝑗2⎞⎟⎟⎟⎟⎠,(3.52) where if 𝛼₁,𝛼₂ are negative, we multiply 𝐮_𝑘1,𝐮_𝑘2 by minus one. Thus, 𝐵=𝐴+𝛼₁𝐮_𝑘1𝐯^∗𝑗₁+𝛼₂𝐮_𝑘2𝐯^∗𝑗₂=𝑈Λ0𝑉^∗,(3.53) where Λ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜎¹⋱𝜎_𝑘1⋯𝛼_𝑘1,𝑗1⋱𝜎⋮_𝑗1⋱𝜎_𝑘2⋯𝛼_𝑘2,𝑗2⋱𝜎⋮_𝑗2⋱𝜎_𝑛⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.54) By permuting rows and columns of Λ it follows that singular values of (^Λ0) are 𝜎₁,…,𝜎_𝑘1−1,𝜎_𝑘1+1,…,𝜎_𝑗1−1,𝜎_𝑗1+1,…,𝜎_𝑘2−1,𝜎_𝑘2+1,…,𝜎_𝑗2−1,𝜎_𝑗2+1,…,𝜎_𝑛(3.55) together with the singular values of 𝐶₁=𝜎^𝑘1𝛼_𝑘1,𝑗10𝜎_𝑗1,𝐶²=𝜎^𝑘2𝛼_𝑘2,𝑗20𝜎_𝑗2.(3.56) Is immediate that the singular values of these matrices are 𝜎_𝑘1=⎛⎜⎜⎜⎜⎝𝛼²¹+𝜎^2𝑘₁+𝜎^2𝑗₁+𝛼²¹+𝜎^2𝑘₁+𝜎^2𝑗₁²−4𝜎^2𝑘₁𝜎^2𝑗₁2⎞⎟⎟⎟⎟⎠^1/2𝜎_𝑗1,=⎛⎜⎜⎜⎜⎝𝛼²¹+𝜎^2𝑘₁+𝜎^2𝑗₁−𝛼²¹+𝜎^2𝑘₁+𝜎^2𝑗₁²−4𝜎^2𝑘₁𝜎^2𝑗₁2⎞⎟⎟⎟⎟⎠^1/2𝜎_𝑘2,=⎛⎜⎜⎜⎜⎝𝛼²²+𝜎^2𝑘₂+𝜎^2𝑗₂+𝛼²²+𝜎^2𝑘₂+𝜎^2𝑗₂²−4𝜎^2𝑘₂𝜎^2𝑗₂2⎞⎟⎟⎟⎟⎠^1/2𝜎_𝑗2,=⎛⎜⎜⎜⎜⎝𝛼²²+𝜎^2𝑘₂+𝜎^2𝑗₂−𝛼²²+𝜎^2𝑘₂+𝜎^2𝑗₂²−4𝜎^2𝑘₂𝜎^2𝑗₂2⎞⎟⎟⎟⎟⎠^1/2.(3.57) The matrix 𝐵=𝐴+𝛼₁𝐮_𝑘1𝐯^∗𝑗₁+𝛼₂𝐮_𝑘2𝐯^∗𝑗₂ can be written as 𝐵Λ==𝑈⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜎₁Λ0𝑉^∗with𝑈=𝑈₁𝑈₂,𝑉=𝑉₁𝑉₂,⋱𝜎_𝑘1𝜎_𝑗1⋱𝜎_𝑘2𝜎_𝑗2⋱𝜎_𝑛⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.58) where the 𝜎’s are not ordered, 𝑈𝑉₁₁=𝐮₁,𝐮₂⋯𝐮_𝑘1,𝐮_𝑗1⋯𝐮_𝑘1+1,𝐮_𝑗1+1⋯𝐮_𝑘2,𝐮_𝑗2⋯𝐮_𝑘2+1,𝐮_𝑗2+1⋯𝐮_𝑚,𝑈₂=𝐯₁,𝐯₂⋯𝐯_𝑘1,𝐯_𝑗1⋯𝐯_𝑘1+1,𝐯_𝑗1+1⋯𝐯_𝑘2,𝐯_𝑗2⋯𝐯_𝑘2+1,𝐯_𝑗2+1⋯𝐯_𝑛,𝑉₂=𝐞₁⋯𝐞_𝑘1−1,𝐮̃_𝑘1𝐮̃_𝑘1+1,𝐞_𝑘1+2⋯𝐞_𝑘2−1,𝐮̃_𝑘2,𝐮̃_𝑘2+1,𝐞_𝑘2+2⋯𝐞_𝑚,=𝐞₁⋯𝐞_𝑘1−1,𝐯̃_𝑘1𝐯̃_𝑘1+1,𝐞_𝑘1+2⋯𝐞_𝑘2−1,𝐯̃_𝑘2,𝐯̃_𝑘2+1,𝐞_𝑘2+2⋯𝐞_𝑛.(3.59) We have proved the following result.

Theorem 3.5. Let A be an 𝑚×𝑛 matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟≥0,𝑟=min{𝑚,𝑛} and SVD 𝐴=𝑈Σ𝑉^∗, where 𝑈=𝐮₁,𝐮₂⋯𝐮_𝑘1⋯𝐮_𝑘2⋯𝐮_𝑚,𝑉=𝐯₁,𝐯₂⋯𝐯_𝑗1⋯𝐯_𝑗2⋯𝐯_𝑛,(3.60)𝑘_𝑖,𝑗_𝑖≤𝑟,𝑘_𝑖≠𝑗_𝑖, are 𝑚×𝑚 and 𝑛×𝑛 unitary matrices, respectively, and Σ=diag{𝜎₁,𝜎₂,…,𝜎_𝑟}. Let 𝛼₁,𝛼₂ be real numbers. Then 𝐵=𝐴+𝛼₁𝐮_𝑘1𝐯^∗𝑗₁+𝛼₂𝐮_𝑘2𝐯^∗𝑗₂ has singular values ^𝑟_{𝑞=1𝑞≠𝑘𝑖,𝑗𝑖}𝜎_𝑞∪𝜎_𝑘1𝜎_𝑘𝑖,𝜎_𝑗1,𝜎_𝑘2,𝜎_𝑗2,(3.61)=⎛⎜⎜⎜⎜⎝𝛼^2𝑖+𝜎^2𝑘_𝑖+𝜎^2𝑗_𝑖+𝛼^2𝑖+𝜎^2𝑘_𝑖+𝜎^2𝑗_𝑖²−4𝜎^2𝑘_i𝜎^2𝑗_𝑖2⎞⎟⎟⎟⎟⎠^1/2𝜎_𝑗𝑖,(3.62)=⎛⎜⎜⎜⎜⎝𝛼^2𝑖+𝜎^2𝑘_𝑖+𝜎^2𝑗_𝑖−𝛼^2𝑖+𝜎^2𝑘_𝑖+𝜎^2𝑗_𝑖²−4𝜎^2𝑘_𝑖𝜎^2𝑗_𝑖2⎞⎟⎟⎟⎟⎠^1/2(3.63)𝑖=1,2. The singular vectors are given by 𝐮̃_𝑞=𝐮_𝑞,𝑞𝐮=̃1,…,𝑚,𝑞≠𝑘_𝑖,𝑞≠𝑗_𝑖,𝑖=1,2,_𝑘1𝐮̃_𝑗1=𝜔⁽¹⁾¹¹𝐮_𝑘1+𝜔⁽¹⁾²¹𝐮_𝑗1,𝐮̃_𝑘2=𝜔⁽¹⁾¹²𝐮_𝑘1+𝜔⁽¹⁾²²𝐮_𝑗1,𝐮̃_𝑗2=𝜔⁽²⁾¹¹𝐮_𝑘2+𝜔⁽²⁾²¹𝐮_𝑗2,=𝜔⁽²⁾¹²𝐮_𝑘2+𝜔⁽²⁾²²𝐮_𝑗2,(3.64) where the coefficients 𝜔’s are obtained as before, and 𝐯_𝑞̃=𝐯_𝑞,𝑞=𝐯̃1,…,𝑛,𝑞≠𝑘_𝑖,𝑞≠𝑗_𝑖,𝑖=1,2,_𝑘1𝐯̃_𝑗1=𝜐⁽¹⁾¹¹𝐯_𝑘1+𝜐⁽¹⁾²¹𝐯_𝑗1,𝐯̃_𝑘2=𝜐⁽¹⁾¹²𝐯_𝑘1+𝜐⁽¹⁾²²𝐯_𝑗1,𝐯̃_𝑗2=𝜐⁽²⁾¹¹𝐯_𝑘2+𝜐⁽²⁾²¹𝐯_𝑗2,=𝜐⁽²⁾¹²𝐯_𝑘2+𝜐⁽²⁾²²𝐯_𝑗2.(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 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}, and with singular value decomposition 𝐴=𝑈Σ𝑉^∗, where 𝑈=𝐮₁∣⋯∣𝐮_𝑝∣⋯Σ∣=𝐮_𝑚dia,g𝑉𝜎=₁𝐯₁∣⋯∣𝐯_𝑞∣⋯∣𝐯_𝑛,𝑝,𝑞≤𝑟,,𝜎₂,…,𝜎_𝑟.(3.66) Then 𝐴+𝛼𝐮_𝑝𝐯^𝑇𝑞 is nonnegative for 𝛼>0 if 𝑎_𝑖𝑘+𝛼(𝐮_𝑝𝐯^𝑇𝑞)_𝑖𝑘≥0,𝑖=1,…,𝑚,𝑘=1,…,𝑛. If (𝐮_𝑝𝐯^𝑇𝑞)_𝑖𝑘<0, then 𝑎_𝑖𝑘+𝛼𝐮_𝑝𝐯^𝑇𝑞_𝑖𝑘≥0iff0<𝛼≤min_𝑖,𝑘𝑎_𝑖𝑘||𝐮_𝑝𝐯^𝑇𝑞_𝑖𝑘||.(3.67) Then we have the following result.

Lemma 3.6. Let 𝐴 be an 𝑚×𝑛 positive matrix with singular values 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0,𝑟=min{𝑚,𝑛}. Let 𝛼 be in the interval 0,min^𝑝,𝑞min_𝑖,𝑘𝑎_𝑖𝑘||𝐮_𝑝𝐯^𝑇𝑞_𝑖𝑘||.(3.68) Then 𝐴+𝛼𝐮_𝑝𝐯^𝑇𝑞,1≤𝑝≤𝑚,1≤𝑞≤𝑛,𝑝≠𝑞, is nonnegative with singular values 𝜎₁,…,𝜎_𝑝−1,𝜎_𝑝,𝜎_𝑝+1,…,𝜎_𝑞−1,𝜎_𝑞,𝜎_𝑞+1,…,𝜎_𝑟,(3.69) where 𝜎_𝑝 and 𝜎_𝑞 are defined as in (3.62) and (3.63), respectively.

Example 3.7. Let 𝐴=(¹²³⁴⁵⁶) be with singular values 𝜎₁=9.5255,𝜎₂=0.5143.𝐴=𝑈Σ𝑉^𝑇, where 𝑈=000...258221949876544−−000.4..82084130947068−000..44.8001886225550,𝑉=00..6718946839−00.7.681498693.(3.70) Then, from (3.68) we have for 𝑝=1,2,3;𝑞=1,2,min_𝑝,𝑞min_𝑖,𝑘𝑎_𝑖𝑘||𝐮_𝑝𝐯^𝑇𝑞_𝑖𝑘||=1.8268(3.71) and 𝛼∈(0,1.8268]. For 𝛼=1.8268, we have 𝐴+𝛼𝐮₁𝐯^𝑇2==135136...371246257925642+1513...7.08437209266828000...146814013483163−−−000...135420257418247(3.72) is nonnegative with singular values 𝜎₁=9.6996 and 𝜎₂=0.5050.

To show that mixed perturbations preserve doubly stochastic structure, observe from Lemma 2.8 that 𝐮_𝑝𝐯^𝑇𝑞𝐞=0 for 𝑞=2,…,𝑛. Then, if 𝐴 is an 𝑛×𝑛 positive doubly stochastic matrix, we have that 𝐴+𝛼𝐮_𝑝𝐯^𝑇𝑞 is doubly stochastic.

Acknowledgment

This work was supported by Fondecyt 1085125, Chile.