## 4 Equations for the Perturbation Parameters
Theorem 3.1: _The Equations (26) and (29) have a unique solution if and only if the singular values of \(A\) are distinct._  
Proof: The Equations (26) and (29) have a unique solution for \(x_{(1)}\) and \(y\) if and only if the square symmetric block matrix  
\[\left[ \begin{array}{cc}-S_{1}&S_{2}\\ S_{2}&-S_{1}\end{array} \right]\]  
is nonsingular, or equivalently, the matrices \(S_{1}\), \(S_{2}\), and \(S_{2}^{2}-S_{1}^{2}\) are nonsingular. The matrices \(S_{1}\) and \(S_{2}\) are nonsingular since the matrix \(A\) has nonzero singular values. (For the solution of linear systems of equations with block matrices, see ([27], Ch. II.) In turn, a condition for nonsingularity of the matrix \(S_{2}^{2}-S_{1}^{2}\) can be found taking into account the structure of the matrices \(S_{xf}\) and \(S_{xg}\). Clearly, the denominators of the first \(n-1\) diagonal entries of \(S_{xf}\) and \(S_{xg}\) will be different from zero if \(\sigma_{1}\) is distinct from \(\sigma_{2},\sigma_{3},\ldots,\sigma_{n}\). Similarly, the denominators of the next group of \(n-2\) diagonal entries will be different from zero if \(\sigma_{2}\) is distinct from \(\sigma_{3},\ldots,\sigma_{n}\), and so on. Finally, \(\sigma_{n-1}\) should be different from \(\sigma_{n}\). Thus, we conclude that the matrices \(S_{xf}\) and \(S_{xg}\) will exist and the Equations (26) and (29) will have a unique solution, if and only if the singular values of \(A\) are distinct. We note that Theorem 3.1 is in accordance with the results obtained in [14; 28]. Such a result should not come as a surprise since \(U\) is the matrix of the transformation of \(AA^{T}\) to Schur (diagonal) form \(U\Sigma\Sigma^{T}U^{T}\) and \(V\) is the matrix of the transformation of \(A^{T}A\) to diagonal form \(V\Sigma^{T}\Sigma V^{T}\). On the other hand, the perturbation problem for the Schur form is well posed only when the matrix eigenvalues (the diagonal elements of \(\Sigma\Sigma^{T}\) or \(\Sigma^{T}\Sigma\)) are distinct. Neglecting the higher-order terms in (31) and (32) and approximating each element of \(f\) and \(g\) by the perturbation norm \(\|\delta A\|_{2}\), we obtain the following result. Lemma 2: _The linear approximations of the vectors \(x_{(1)}\) and \(y\) satisfy_  
\[|x_{(1)}| \preceq x_{(1)}^{lin},\ \ \ |y|\preceq y^{lin} \tag{35}\] \[x_{(1)}^{lin} = (|S_{xf}|+|S_{xg}|)h,\] (36) \[y^{lin} = (|S_{yf}|+|S_{yg}|)h, \tag{37}\]  
_where \(S_{yg}=S_{xf}\) and \(S_{yf}=S_{xg}\) are given by (33) and (34), respectively, and_  
\[h=\underbrace{[1,1,\ldots,1]}_{g}{}^{T}\times\|\delta A\|_{2}.\]Clearly, if the matrices \(|S_{xf}|,\ |S_{gg}|,\ |S_{yy}|,\ |S_{gg}|\) have large diagonal entries, then the estimates of the perturbation parameters will be large. Using the expressions for \(S_{xf}\) and \(S_{xg}\), we may show that  
\[\|S_{xf}\|_{2} = \max_{i,j}\frac{\sigma_{i}}{|\sigma_{j}^{2}-\sigma_{i}^{2}|},i=1,2, \ldots,n-1,\ j=i+1,i+2,\ldots,n,\] \[\|S_{xg}\|_{2} = \max_{i,j}\frac{\sigma_{j}}{|\sigma_{j}^{2}-\sigma_{i}^{2}|}.\]  
Note that the norms of \(S_{xf}\) and \(S_{xg}\) can be considered as condition numbers of the vectors \(x_{1}\) and \(y\) with respect to the changes in \(\delta A\). An asymptotic estimate of the vector \(x_{(2)}\) is obtained, neglecting the higher-order term \(\Delta_{2}\) and approximating its elements according to (30). The linear approximation of the vector \(x_{(2)}\) satisfies  
\[x_{(2)}^{lin} = \operatorname{vec}(Z), \tag{38}\] \[Z = \|\delta A\|_{2}\times\left[ \begin{array}{cccc}1/\sigma_{1}&1/ \sigma_{2}&\ldots&1/\sigma_{n}\\ 1/\sigma_{1}&1/\sigma_{2}&\ldots&1/\sigma_{n}\\ \vdots&\vdots&\vdots&\vdots\\ 1/\sigma_{1}&1/\sigma_{2}&\ldots&1/\sigma_{n}\end{array} \right].\]  
Equation (38) shows that a group of \(n\) elements of \(x_{(2)}^{lin}\) will be large if the singular value associated with the corresponding column of \(Z\) is small. The presence of large elements in the vector \(x\) leads to large entries in \(\delta W_{U}\) and consequently in the estimate of \(\delta U\).