## 4 Equations for the Perturbation Parameters  
In this section, we derive exact nonlinear equations for the perturbation parameters \(x\) and \(y\) and equations for their linear approximations. At this stage, we assume that the perturbation \(\delta A\) is known, but in Section 5 we show how to use these equations to find asymptotic approximations of \(x\) and \(y\) knowing only the perturbation norm. The elements of the perturbation parameter vectors \(x\) and \(y\) can be determined from Equation (5). For this aim, it is appropriate to transform this equation as follows. Taking into account that \(AV=U\Sigma=U_{1}\Sigma_{n}\) and \(U^{T}A=\Sigma V^{T}\), the equation is represented as  
\[\delta U^{T}U_{1}\Sigma_{n}+\left[ \begin{array}{c}\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]V^{T}\delta V+\delta F=\left[ \begin{array} []{c}\widetilde{\Sigma}_{n}-\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]+\Delta_{0}. \tag{17}\]where  
\[\delta F=\left[ \begin{array}{c|cccc}\hline\delta I_{1}&\delta I_{2}&\ldots& \delta I_{1n}\\ \hline\delta I_{2}&\delta I_{22}&\ldots&\delta I_{2n}\\ \hline\vdots&\vdots&\ddots&\vdots\\ \hline\delta I_{n1}&\delta I_{n2}&\ldots&\delta I_{nn}\\ \hline\delta I_{n\pi+1,1}&\delta I_{n\pi+1,2}&\ldots&\delta I_{n\pi+1,n}\\ \vdots&\vdots&\ddots&\vdots\\ \hline\delta I_{m1}&\delta I_{m2}&\ldots&\delta I_{mn}\\ \end{array} \right]:=\left[ \begin{array}{c}\delta F_{1}\\ \hline\delta F_{2}\\ \end{array} \right]\text{, }\delta F_{1}\in\mathbb{R}^{n\times n}\text{, }\delta F_{2}\in \mathbb{R}^{(m-n)\times n}\text{. }\]  
After transposing (9), we obtain that  
\[\delta U^{T}U_{1}=-U^{T}\delta U_{1}-\delta U^{T}\delta U_{1}. \tag{18}\]  
Substituting in (17) the term \(\delta U^{T}U_{1}\) with the expression on the right hand side of (18), we obtain  
\[-U^{T}\delta U_{1}\Sigma_{n}+\left[ \begin{array}{c}\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]V^{T}\delta V+\delta F=\left[ \begin{array} []{c}\widetilde{\Sigma}_{n}-\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]+\Delta, \tag{19}\]  
where  
\[\Delta = \Delta_{0}+\delta U^{T}\delta U_{1}\Sigma_{n} \tag{20}\] \[= \delta U^{T}\delta U_{1}\Sigma_{n}-\delta U^{T}A\delta V-U^{T} \delta A\delta V-\delta U^{T}\delta AV-\delta U^{T}\delta A\delta V\]  
contains higher-order terms in the entries of \(\delta A\), \(\delta U\), and \(\delta V\). Replacing the matrices \(U^{T}\delta U_{1}\) and \(V^{T}\delta V\) by \(\delta W_{U}\), and \(\delta W_{V}\), respectively, (19) is rewritten as  
\[-\delta W_{U}\Sigma_{n}+\left[ \begin{array}{c}\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]\delta W_{V}+\delta F=\left[ \begin{array} []{c}\widetilde{\Sigma}_{n}-\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]+\Delta\]  
or  
\[-\delta Q_{U}\Sigma_{n} + \left[ \begin{array}{c}\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]\delta Q_{V}+\delta F=\left[ \begin{array} []{c}\widetilde{\Sigma}_{n}-\Sigma_{n}\\ 0_{(m-n)\times n}\end{array} \right]\] \[+(\delta D_{U}-\delta N_{U})\Sigma_{n}-\Sigma(\delta D_{V}-\delta N _{V})+\Delta.\]  
Note that the matrices \(\delta D_{U},\delta N_{U},\delta D_{V},\delta N_{V}\) and \(\Delta\) contain only higher-order terms in the entries of \(\delta A\), \(\delta U\) and \(\delta V\). Equation (21) is the basic equation in the perturbation analysis of the SVD performed in this paper. This equation represents a diagonal system of linear equations with respect to the unknown entries of \(\delta Q_{U}\) and \(\delta Q_{V}\), allowing us to solve it efficiently even for high-order matrices. Neglecting the higher-order terms in this equation, we obtain in Section 5 asymptotic bounds for the elements of the SVD. The approximation of these terms makes it possible to determine global perturbation bounds in Section 9. The entries of the matrices \(\delta Q_{U}\) and \(\delta Q_{V}\) can be substituted by the corresponding elements \(x_{k}\), \(k=i+(j-1)m-\frac{i(j+1)}{2}\) and \(y_{\ell}\), \(\ell=i+(j-1)n-\frac{i(j+1)}{2}\) of the vectors \(x\) and \(y\) as shown in the previous section.