Nonnegativity Preservation under Singular Values Perturbation
We study how singular values and singular vectors of a matrix change, under matrix perturbations of the form and , , where , is an positive matrix with singular values , and , 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.
A singular value decomposition of a matrix is a factorization where and both and are unitary. The diagonal entries of are called the singular values of The columns of are called left singular vectors of and the columns of are called right singular vectors of Every has a singular value decomposition and the following relations hold: and If then and may be taken to be real (see ).
Let be an positive matrix with singular values and left and right singular vectors respectively. In this paper we study how singular values and singular vectors of change, under matrix perturbations of the form and Perturbations of the form were used in  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 ). 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 . 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 , 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  to modify eigenvalues of a matrix of order via a perturbation of rank- without changing any of the 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 ). In  the following simple singular value version of the Rado and Brauer results were given.
Theorem 1.1. Let be an matrix with singular values Let be matrices of order and whose columns are the left and right singular vectors, respectively, corresponding to . Let with Then has singular values
Note that the singular values of are not necessarily in nondecreasing order. However we can reorderer them by using an appropriate permutation.
Corollary 1.2. Let be an matrix with singular values Let and respectively, the left and right singular vectors corresponding to . Let such that Then has singular values
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 then we may choose and in such a way that
The paper is organized as follows. In Section 2 we consider perturbations of the form 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 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 be an positive matrix with singular values In this section we consider perturbations which preserve nonnegativity. Note that if is an nonnegative matrix, then the left and right singular vectors and corresponding to the maximal singular value respectively, are nonnegative. Hence, in this case, the matrix is nonnegative for all .
Now, let us consider the perturbation with . Let and be the left and right singular vectors corresponding to respectively. Let and let the entry in position of be negative. That is, Then if is nonnegative, Thus, to preserve the nonnegativity of it is enough to choose in the interval provided that otherwise must be zero. Then from (2.2) and Theorem 1.1 we have the following result.
Lemma 2.1. Let be an positive matrix with singular values Let be in the interval Then is nonnegative with singular values
Remark 2.2. It is clear that if in Lemma 2.1, is taken in then is positive with singular values Moreover, for in intervals in Lemma 2.1 and this remark, the nonnegativity is obtained independently of the chosen singular vectors
A more handle interval for is given by the following lemma.
Lemma 2.3. Let be an positive matrix with singular values Let be in the interval Then is nonnegative with singular values
Now we consider rank- perturbations, where That is, perturbations of the form as in Theorem 1.1. Let be an positive matrix with singular values Then will be nonnegative if From the family of straight lines it follows that they intersect the axes and at points where and respectively. Let Let Then is nonnegative for
A more handle region for is given by the following lemma.
Lemma 2.5. Let be an positive matrix with singular values Let be the intersection points in (2.17). Let where is the norm. Then is nonnegative with singular values
Remark 2.7. Let be an complex matrix with singular values and singular value decomposition In  it was defined the concept of energy of as If is positive, then as an application of the rank- perturbation result, from Lemma 2.5 and Remark 2.4, we may construct, for a family of nonnegative matrices with Now, suppose is nonnegative with where is an upper bound. Then from (1.2), by taking we may construct a family of nonnegative matrices with
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 matrix is said to be with constant row sums if We denote by the set of all matrices with constant row sums equal to
Lemma 2.8. Let be an irreducible doubly stochastic matrix and let with Then
Proof. Since is doubly stochastic, then and Then, and since
The following result shows that simple perturbations preserve doubly stochastic structure.
Proposition 2.9. Let be an irreducible doubly stochastic matrix. Then,
Proof. Since then and Hence, the singular vectors and are Then and Thus (i) holds. From Lemma 2.8, and (ii) holds. From (i) and (ii) we have (iii)
Example 2.10. Consider the matrix which is nonnegative generalized doubly stochastic, that is, is nonnegative with and has singular values Let be the singular value decomposition of Let Then is nonnegative generalized doubly stochastic with singular values
3. Nonnegativity Preservation under Mixed Perturbations
In this section we discuss matrix perturbations of the form with 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 be a matrix with singular values Let with and Let That is, The matrix has the singular value decomposition: Then, Now we compute the singular values of the matrix by computing the eigenvalues of where Since then with being the eigenvalues of Thus, we obtain Hence, the singular values of are .
Now we generalize these results for matrices.
Theorem 3.1. Let be an matrix with singular values and with singular value decomposition , where Let Then has singular values, where
Proof. Without loss of generality we may assume that and . The matrix has a singular value decomposition where The decomposition in (3.12) can be written as where and are and permutation matrices of the form with Since where if we multiply the vector by minus one. Then where By applying row and column permutations on the matrix , it follows from (3.17) and (3.18) that the singular values of are where and are as in (3.10) and (3.11), respectively.
Observe that in Theorem 3.1, has singular values if . If then only corresponding to the right singular vector changes and take the form A straightforward calculation shows that for Observe that
Example 3.2. Let with singular values Let left and right singular vectors of respectively. Let Then, from (3.10) and (3.11), the matrix has singular values and By using Theorem 1.1 we have that has singular values
Different from perturbations of the form the perturbation of Theorem 3.1 affects not only the singular values and but also the corresponding left and right singular vectors and To make this modification clear, we consider again the previous discussion to Theorem 3.1: Let the SVD of with obtained from (3.10) and (3.11), respectively. The left singular vectors of (eigenvectors of ) are and its corresponding right singular vectors (eigenvectors of ) are Thus, can be written as where Now, we generalize this result. Without loss generality we assume that . From (3.17) and (3.18), by permuting rows and columns of the matrix , we have The singular vectors of are obtained from the singular vectors of the matrix Let Then the unitary matrices of the singular value decomposition of are of order and , respectively. Then where and are not ordered. Since then the singular vectors corresponding to have been modified. We have prove the following result.
Corollary 3.3. Let be an matrix with singular values and with singular value decomposition , where Let Then has left singular vectors, and right singular vectors
Observe that if orthogonal matrices in (3.32) are of the same type then a straight forward calculation shows that (3.37) and (3.38) become and respectively, while if they are of different type, then (3.38) becomes From (3.10) and (3.11) it is clear that with . The following result tells us how the new singular values relate with the previous singular values .
Proof. Since we have Thus In the same way, Then Therefore, Observe that In particular for and , all singular values of are in the interval .
Now we extend the mixed perturbation result given by Theorem 3.1 to rank-2 perturbations, that is, perturbations of the form , with nonzero real numbers and being left and right singular vectors of respectively. Then as in (3.12) with It is clear that the matrix in (3.49) can be written as where are permutation matrices of the form From (3.50), the matrix in (3.49) is where if are negative, we multiply by minus one. Thus, where By permuting rows and columns of it follows that singular values of are together with the singular values of Is immediate that the singular values of these matrices are The matrix can be written as where the ’s are not ordered, We have proved the following result.
Theorem 3.5. Let be an matrix with singular values and SVD where are and unitary matrices, respectively, and Let be real numbers. Then has singular values The singular vectors are given by where the coefficients ’s are obtained as before, and