## 8 Sensitivity of Singular Subspaces
\begin{array}{ccccc}*&*&*&\dots&*\\ x_{1}&*&*&\dots&*\\ x_{2}&x_{m}&*&\dots&*\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{n-1}&x_{m+n-3}&x_{2m+n-6}&\dots&*\\ \vdots&\vdots&\vdots&\vdots\\ x_{m-1}&x_{2m-3}&x_{3m-6}&\dots&x_{p}\end{array} \right]\in\mathbb{R}^{m\times n},\]  
we have that the exact maximum angle between the unperturbed and perturbed left singular subspace of dimension \(r\) is given by  
\[\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r})=\arcsin(\sigma_{\max} (\delta W_{Ur+1:m,1:r})). \tag{47}\]  
Similarly, the maximum angle between the unperturbed and perturbed right singular subspace of dimension \(r\) is  
\[\Theta\max(\mathcal{V}_{r},\widetilde{\mathcal{V}}_{r})=\arcsin(\sigma_{\max }(\delta W_{Vr+1:m,1:r})). \tag{48}\]  
Thus, we obtain the following result. **Theorem 3**. : _The asymptotic estimate of the angle \(\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r})\) between the unperturbed and perturbed singular subspace of dimension \(r\) satisfies_  
\[|\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r})| \leq \Phi\max^{lin}(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}), \tag{49}\] \[\Phi\max^{lin}(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}) = \arcsin(\|\delta W_{Ur+1:m,1:r}^{lin}\|_{2}),\]  
Figure 2: Sensitivity estimations of the right singular subspaces. _where_  
\[\delta W_{U}^{lin}=\left[ \begin{array}{cccccc}0&*&*&\ldots&*\\ x_{1}^{lin}&0&*&\ldots&*\\ x_{2}^{lin}&x_{m}^{lin}&0&\ldots&*\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{n-1}^{lin}&x_{m+n-3}^{lin}&x_{2m+n-6}^{lin}&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ x_{m-1}^{lin}&x_{2m-3}^{lin}&x_{3m-6}^{lin}&\ldots&x_{p}^{lin}\end{array}  \right]\in\mathbb{R}^{m\times n}\]  
_and the parameters \(x_{k}^{lin},\ k=1,2,\ldots,p\) are determined from (39)._  
In particular, for the sensitivity of the range \(\mathcal{R}(A)\) of \(A\), we obtain that  
\[\sin(\Phi\max^{lin}(\mathcal{U}_{n},\widetilde{\mathcal{U}}_{n}))=\|\delta W _{U_{n+1:m,1:n}}\|_{2}.\]  
Similarly, for the angles between the unperturbed and perturbed right singular subspaces of dimension \(r\), we obtain the linear estimates  
\[ \begin{array}{rcl}|\Theta\max(\mathcal{V}_{r},\widetilde{\mathcal{V}}_{r}) |&\leq&\Theta\max^{lin}(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}),\\ \Theta\max^{lin}(\mathcal{V}_{r},\widetilde{\mathcal{V}}_{r})&=&\arcsin(\| \delta W_{V_{r+1:n,1:r}}^{lin}\|_{2}),\end{array}  \tag{50}\]  
where  
\[\delta W_{V}^{lin}=\left[ \begin{array}{cccccc}0&*&*&\ldots&*\\ y_{1}^{lin}&0&*&\ldots&*\\ y_{2}^{lin}&y_{n}^{lin}&0&\ldots&*\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ y_{n-1}^{lin}&y_{2n-3}^{lin}&y_{3n-6}^{lin}&\ldots&0\end{array} \right]\in \mathbb{R}^{n\times n}\]  
and \(y_{\ell},\ \ell=1,2,\ldots,q\) are determined from (37). We note that using separate \(x\) and \(y\) parameters decouples the SVD perturbation problem and makes it possible to determine the sensitivity estimates of the left and right singular subspaces independently. This is important when the left or right subspace in a pair of singular subspaces is much more sensitive than its counterpart. Consider as an example the same perturbed matrix \(A\) as in Example 1. Computing the matrices \(\delta W_{U}^{lin}\) and \(\delta W_{V}^{lin}\), it is possible to estimate the sensitivity of all four pairs of singular subspaces of dimensions \(r=1,2,3\), and \(4\) corresponding to the chosen ordering of the singular values. In Table 4, we show the actual values of the left and right singular subspaces sensitivity and the computed asymptotic estimates (49) and (50) of this sensitivity. To determine the sensitivity of other singular subspaces, reordering the singular values in the initial decomposition so that the desired subspace appears in the set of nested singular subspaces is possible.