## 3 Perturbed Orthogonal Matrices and Perturbation Parameters
\begin{array}{c}0\\ \vdots\\ 0\\ \delta d_{ujj}\\ 0\\ \vdots\\ 0\end{array} \right]\ \leftarrow j\,\ j=1,2,\ldots,n.\]  
Hence,  
\[\|\delta w_{uj}\|^{2}=\|\delta q_{uj}+\delta d_{ujj}\|_{2}^{2}=\|\delta q_{uj} \|_{2}^{2}+\delta d_{ujj}^{2}. \tag{12}\]  
From (12) and (11), we obtain the quadratic equation  
\[\delta d_{ujj}^{2}+2\delta d_{ujj}+\|\delta q_{uj}\|_{2}^{2}=0. \tag{13}\]  
From the two possible solutions to this equation, we take the root  
\[\delta d_{ujj}=-\|\delta q_{uj}\|_{2}^{2}/(1+\sqrt{1-\|\delta q_{uj}\|_{2}^{2} }),\ j=1,2,\ldots,n, \tag{14}\]  
since in this case \(\delta d_{ujj}\to 0\) with \(\|\delta q_{uj}\|_{2}\to 0\). The expression (14) allows us to find an approximation of \(\delta d_{ujj}\) from the entries of the matrix \(\delta Q_{U}\). For small perturbation \(\delta A\) (small values of \(\|\delta q_{uj}\|_{2}\)), we have the estimate (also following from (11))  
\[|\delta d_{ujj}|\leq\delta d_{ujj}^{\ \ lin}=\|\delta q_{uj}\|_{2}^{2}/2.\]  
So, for small perturbations the quantity \(|\delta d_{ujj}|,\ j=1,2,\ldots,n\) depends quadratically on \(\|\delta q_{uj}\|_{2}\). Thus, the matrix \(\delta W_{U}\) can be represented as the sum  
\[\delta W_{U}=\delta Q_{U}+\delta D_{U}-\delta N_{U}, \tag{15}\]where according to (8), the matrix \(\delta Q_{U}\) has entries depending only on the perturbation parameters \(x\),  
\[\delta D_{U}=\left[ \begin{array}{cccc}\underline{\delta d_{U+1}}&0&\ldots&0\\ \hline\underline{\delta d_{U+2}}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\underline{\delta d_{U+n}}\\ \hline 0&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\ldots&0\end{array} \right]\in\mathbb{R}^{m\times n},\]  
and the matrix  
\[\delta N_{U}=\left[ \begin{array}{cccc}0&\underline{\delta u}\cdot\underline {\delta u}&\underline{\delta u}\cdot\underline{\delta u}&\ldots&\ldots&\ldots \\ 0&0&\ldots&\ldots&\ldots&\ldots\\ 0&0&0&\ldots&\ldots&\ldots\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&\ldots&\ldots\\ 0&0&0&\ldots&0\\ \hline 0&0&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\ldots&0\end{array} \right]\in\mathbb{R}^{m\times n}\]  
contains second-order terms in \(\delta u_{j},\ j=1,2,\ldots,n\). Since the matrices \(\delta D_{U}\) and \(\delta N_{U}\) contain only second-order terms in the entries of \(\delta U\), taking into account that \(x\to 0\) and consequently \(\delta U\to 0\) with \(\|\delta A\|_{2}\to 0\), it follows from (15) that the matrix \(\delta Q_{U}\) is the linear approximation of \(\delta W_{U}\). Similarly, for the matrix  
\[\delta W_{V}=V^{T}\delta V = \left[ \begin{array}{cccc}\underline{\delta u}\cdot\underline {\delta u}&\underline{\delta u}\cdot\underline{\delta u}&\ldots&\ldots&\ldots \\ \underline{\delta u}\cdot\underline{\delta u}&\underline{\delta u}\cdot \underline{\delta u}&\ldots&\ldots&\ldots\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \underline{\delta u}\cdot\underline{\delta u}&\underline{\delta u}\cdot \underline{\delta u}&\ldots&\ldots&\ldots\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ \underline{\delta u}\cdot\underline{\delta u}&\underline{\delta u}\cdot \underline{\delta u}&\ldots&\ldots&\ldots\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&0\end{array} \right]\in\mathbb{R}^{n\times n},\]  
like the case of \(\delta W_{U}\), it is possible to show that  
\[\delta W_{V}=\delta Q_{V}+\delta D_{V}-\delta N_{V}, \tag{16}\]where  
\[\delta Q_{V} = \left[ \begin{array}{ccccc}\textbf{0}&\textbf{0}&\textbf{-}\textbf{ \frac{1}{2}}&\textbf{-}\textbf{\frac{1}{2}}&\cdots&\textbf{-}\textbf{\frac{1}{2 }}\\ \textbf{0}&\textbf{0}&\textbf{-}\textbf{\frac{1}{2}}&\cdots&\textbf{-}\textbf{ \frac{1}{2}}&\cdots\\ \textbf{0}&\textbf{0}&\textbf{-}\textbf{\frac{1}{2}}&\textbf{-}\textbf{\frac{1 }{2}}&\cdots&\textbf{-}\textbf{\frac{1}{2}}&\cdots\\ \textbf{0}&\textbf{0}&\textbf{-}\textbf{\frac{1}{2}}&\cdots&\textbf{-}\textbf{ \frac{1}{2}}&\cdots&\textbf{-}\textbf{\frac{1}{2}}\\ \end{array}