## 1 Introduction  
As it is known ([1], Ch. 2), ([2], Ch. 1), ([3], Ch, 2), each \(m\times n\), \(m\geq n\) matrix \(A\in\mathbb{R}^{m\times n}\) can be represented by the _singular value decomposition_ (SVD) in the factorized form  
\[A=U\Sigma V^{T}, \tag{1}\]  
where the \(m\times m\) matrix \(U\) and the \(n\times n\) matrix \(V\) are orthogonal and the \(m\times n\) matrix \(\Sigma\) is diagonal:  
\[\Sigma=\left[ \begin{array}{c}\Sigma_{n}\\ 0_{(m-n)\times n)}\end{array} \right],\;\;\Sigma_{n}=\mathrm{diag}(\sigma_{1}, \sigma_{2},\ldots,\sigma_{n}).\]  
The numbers \(\sigma_{i}\geq 0\) are called _singular values_ of the matrix \(A\). The columns of  
\[U:=[u_{1},u_{2},\ldots,u_{m}],\;u_{j}\in\mathbb{R}^{m},\]  
are called _left singular vectors_ and the columns of  
\[V:=[v_{1},v_{2},\ldots,v_{n}],\;v_{j}\in\mathbb{R}^{n}\]  
are the _right singular vectors_. The subspaces spanned by sets of left and right singular vectors are called _left and right singular subspaces_, respectively. The singular value decomposition has a long and interesting history described in [4]. The SVD has many properties, making it an invaluable tool in matrix analysis and matrix computations; see the references cited above. Among them is that the rank \(r\) of equals the number of its nonzero singular values and the equality \(\|A\|_{2}=\sigma_{\max}\). The usual assumption is that by an appropriate ordering of the columns \(u_{j}\) and \(v_{j}\), the singular values appear in the order  
\[\sigma_{\max}:=\sigma_{1}\geq\sigma_{2}\geq\ldots\geq\sigma_{n}:=\sigma_{\min}.\]  
If some of the singular values are equal to zero, then \(A\) has \(r<n\) linearly independent columns, and the matrix \(\Sigma\) in (1) can be represented as  
\[\Sigma=\left[ \begin{array}{cc}\Sigma_{r}&0_{r\times(n-r)}\\ 0_{(m-r)\times r}&0_{(m-r)\times(n-r)}\end{array} \right],\ \ \Sigma_{r}=\operatorname{diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{r}).\]  
Further, we shall consider the case \(r=n\), i.e., assume that the matrix \(A\) is of full column rank. This paper concerns the case when the matrix \(A\) is subject to an additive perturbation \(\delta A\). In such a case, there exists another pair of orthogonal matrices \(\widetilde{U}\) and \(\widetilde{V}\) and a diagonal matrix \(\widetilde{\Sigma}\), such that  
\[\widetilde{A}:=A+\delta A=\widetilde{U}\widetilde{\Sigma}\widetilde{V}^{T}. \tag{2}\]  
The perturbation analysis of the singular value decomposition consists in determining the changes in the quantities related to the elements of the decomposition due to the perturbation \(\delta A\). This includes determining bounds on the changes of the entries of the orthogonal matrices that reduce the original matrix to diagonal form and bounds on the perturbations of the singular values. Hence, the analysis aims to find bounds on the sizes of \(\delta U=\widetilde{U}-U\), \(\delta V=\widetilde{V}-V\), and \(\delta\Sigma=\widetilde{\Sigma}-\Sigma\) as functions of the size of \(\delta A\). It should be emphasized that problems of such a kind arise in most singular value decomposition applications. The most important application of the perturbation analysis is to assess the accuracy of the computed SVD since each algorithm for its determination produces the SVD of \(A+\delta A\), not of \(A\), where the size of \(\delta A\) depends on the properties of the floating point arithmetic used and the corresponding algorithm. Knowing the size of \(\delta A\), we may use the perturbation bounds to estimate the difference between the actual and computed elements of the decomposition. For a deep and systematic presentation of the matrix perturbation theory and its use in accuracy estimation, the reader is referred to the books of [PERSON] [2], [PERSON] and [PERSON] [5], and [PERSON] [6]. According to [PERSON]'s theorem ([2], Ch.