## 8 Sensitivity of Singular Subspaces  
The sensitivity of the left  
\[\mathcal{U}_{r}=\text{span}(u_{1},u_{2},\ldots,u_{r}),r\leq\min\{m-1,n\}\]  
or the right  
\[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]  
singular subspace of dimension \(r\) is measured by the canonical angles between the corresponding unperturbed and perturbed subspaces ([2], Ch. 4), ([5], Ch. V), [30]. Let the unperturbed left singular subspace corresponding to the first \(r\) singular values be denoted by \(\mathcal{U}_{r}\) and its perturbed counterpart as \(\widetilde{\mathcal{U}}_{r}\), and let \(U_{(r)}\) and \(\widetilde{U}_{(r)}\) be the orthonormal bases for \(\mathcal{U}_{r}\) and \(\widetilde{\mathcal{U}}_{r}\), respectively. Further on, the sensitivity of the singular subspace \(\mathcal{U}_{r}\) will be characterized by the maximum canonical angle between \(\mathcal{U}_{r}\) and \(\widetilde{\mathcal{U}}_{r}\), defined as  
\[\cos(\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}))=\sigma_{\min}(U_{ (r)}^{T}\widetilde{U}_{(r)}). \tag{44}\]  
The expression (44) has the disadvantage that if \(\Phi\max\) is small, then \(\cos(\Phi\max)\approx 1\) and the angle \(\Phi\max\) is not well determined. To avoid this difficulty, instead of \(\cos(\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}))\), it is preferable to work with \(\sin(\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}))\). Let \(U_{(r)}^{\perp}\) be the orthogonal complement of \(U_{(r)}\), \({U_{(r)}^{\perp}}^{T}U_{(r)}=0\). Then, it is possible to show that [31]  
\[\sin(\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}))=\sigma_{\max}({U_ {(r)}^{\perp}}^{T}\widetilde{U}_{(r)}). \tag{45}\]  
Since  
\[\widetilde{U}_{(r)}=U_{(r)}+\delta U_{(r)},\]  
we have that  
\[\sin(\Phi\max(\mathcal{U}_{r},\widetilde{\mathcal{U}}_{r}))=\sigma_{\max}({U_ {(r)}^{\perp}}^{T}\delta U_{(r)}). \tag{46}\]  
Equation (46) shows that the sensitivity of the left singular subspace \(\mathcal{U}_{r}\) is related to the values of the perturbation parameters \(x_{k}=u_{i}^{T}\delta u_{j},\ k=i+(j-1)m-\frac{i(j+1)}{2},i>r\), \(j=1,2,\ldots,r\). In particular, for \(r=1\), the sensitivity of the first column of \(U\) (the left singular vector, corresponding to \(\sigma_{1}\)) is determined by  
\[\sin(\Phi\max(\mathcal{U}_{1},\widetilde{\mathcal{U}}_{1}))=\sigma_{\max}( \delta W_{U2m,1}),\]  
for \(r=2\) one has  
\[\sin(\Phi\max(\mathcal{U}_{2},\widetilde{\mathcal{U}}_{2}))=\sigma_{\max}( \delta W_{U3:m,1:2})\]  
and so on (see Figure 1), where the matrices \({U_{(r)}^{\perp}}^{T}\delta U_{(r)}=\delta W_{Ur+1:m,1:r}\) for different values of \(r\) are highlighted in boxes. Similarly, utilizing the matrix \(\delta W_{V}\), it is possible to find the sine of the maximum angle between the unperturbed \(\mathcal{V}_{r}\) and the perturbed \(\widetilde{\mathcal{V}}_{r}\) right singular subspace  
\[\sin(\Theta\max(\mathcal{V}_{r},\widetilde{\mathcal{V}}_{r}))=\sigma_{\max}( \delta W_{Vr+1:n,1:r})\]  
(see Figure 2). Hence, if the perturbation parameters are determined, it is possible to find sensitivity estimates of the nested singular subspaces  
\[\mathcal{U}_{1}=\mathrm{span}(U_{(1)}) = \mathrm{span}(u_{1}),\] \[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]0 \[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]1 \[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]2  
\[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]3 Sensitivity estimations of the left singular subspaces. and  
\[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]4 \[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]5 \[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]6 \[\mathcal{V}_{r}=\text{span}(v_{1},v_{2},\ldots,v_{r}),\ \ r\leq n-1\]7  
Specifically, as  
\[\delta W_{U}=\left[