## 1 Introduction
1), we have that  
\[|\delta\sigma_{i}|=|\widetilde{\sigma}_{i}-\sigma_{i}|\leq\|\delta A\|_{2},\ i=1,2, \ldots,n, \tag{3}\]  
which shows that the singular values are perturbed by no more than the 2-norm of the perturbation of \(A\); i.e., the singular values are always well conditioned. The SVD perturbation analysis is well defined if the matrix \(A\) is of full column rank \(n\); i.e., \(\sigma_{min}\
eq 0\), since otherwise, the corresponding left singular vector is undetermined. The size of the perturbations \(\delta A\), \(\delta U\), \(\delta V\), and \(\delta\Sigma\) is usually measured by using some of the matrix norms, which leads to the so-called _normwise perturbation analysis_. In several cases, we are interested in the size of the perturbations of the individual entries of \(\delta U\), \(\delta V\), and \(\delta\Sigma\), so that it is necessary to implement a _componentwise perturbation analysis_[7]. This analysis has an advantage when the individual components of \(\delta U\), \(\delta V\), and \(\delta\Sigma\) differ significantly in magnitude, and the normwise estimates do not produce tight bounds on the perturbations. The literature on the perturbation theory of the singular value decomposition is significant. The first results on this topic are obtained by [PERSON] [8] and [PERSON] [9], who developed estimates of the sensitivity of pairs of singular subspaces (see also ([5], Ch. V)). Other results concerning the sensitivity of the singular vectors and singular subspaces are obtained in [10,11], and a survey on the perturbation theory of the SVD until 1990 can be found in [12]. Using the technique of perturbation expansions for invariant subspaces, [PERSON] [13] derived perturbation bounds for a pair of singular subspaces that improved the bounds obtained in [9] and contained as a special case the bounds presented in [11]. A perturbation theory for the singular values and singular subspaces of diagonally dominant matrices, including graded matrices, is presented in [14], and optimal perturbation bounds for the case of structured perturbations are derived in [15]. The seminal paper by [PERSON] and his co-authors [16] contains a perturbation theory that provides relative perturbation bounds for the singular values and singular subspaces for different classes of matrices. The problem of backward error analysis of SVD algorithms is discussed in [17]. High-accuracy algorithms for computing SVD are proposed by [PERSON] and [PERSON] in [18; 19] and implemented in the LAPACK package [20]. An improvement in the results of [8] is presented in [21]. Several results concerning the sensitivity of the SVD are summarized in the survey [22], and a reach bibliography on the accurate computation of SVD can be found in [23]. Finally, some recent applications of SVD are described in [24; 25]. It should be pointed out that the available SVD perturbation theory provides bounds on the sensitivity of the singular vectors and singular subspaces but does not provide perturbation bounds on the individual entries of the matrices \(U\) and \(V\). Such bounds are important in several applications, and this fact justifies the opinion that apart from the large number of results about the sensitivity of the singular values and singular vectors, a complete componentwise perturbation analysis of the SVD is not available up to the moment. This paper presents a rigorous perturbation analysis of the orthogonal matrices, singular subspaces, and singular values of a real matrix of full column rank. It is proved that the SVD perturbation problem is well posed only in the case of distinct (simple) singular values. The analysis produces asymptotic (local) componentwise perturbation bounds of the entries of the orthogonal matrices \(U\) and \(V\) and the singular values of the given matrix. Local bounds are derived for the sensitivity of a pair of singular subspaces measured by the angles between the unperturbed and perturbed subspaces.