## Appendix A Proof of Theorem 1
In the following, we derive the structures of \((\mathbf{P}^{\star},\mathbf{\tilde{T}}^{\star},\mathbf{\tilde{R}}^{\star}, \mathbf{\tilde{Q}}^{\star})\) using Equations (13a)-(13k). Multiplying (13f) on the right-hand side by \((\mathbf{\hat{R}}+\mathbf{\tilde{R}}^{\star})\) and using the condition \((\mathbf{\hat{R}}+\mathbf{\tilde{R}}^{\star})\mathbf{\Gamma}_{R}=\mathbf{0}\) in (13h), we obtain  
\[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]7To satisfy the condition (A4), \(\tilde{\mathbf{R}}^{\star}\) should have the following form: \(\tilde{\mathbf{R}}^{\star}=\frac{\mathrm{Tr}(\mathbf{Z}^{-1})}{\mu_{R}}\mathbf{I }_{M_{\ell}}\) or \(\tilde{\mathbf{R}}^{\star}=-\tilde{\mathbf{R}}\). The latter choice cannot be a solution since it is infeasible when \(||\tilde{\mathbf{R}}||_{F}^{2}>\epsilon_{R}\). Therefore, we have \(\tilde{\mathbf{R}}^{\star}=\frac{\mathrm{Tr}(\mathbf{Z}^{-1})}{\mu_{R}} \mathbf{I}_{M_{\ell}}\). From the constraint \(||\tilde{\mathbf{R}}^{\star}||_{F}^{2}\leq\epsilon_{R}\), the optimal value of \(\mu_{R}\) can be computed as \(\mu_{R}=\mathrm{Tr}(\mathbf{Z}^{-1})\sqrt{\frac{M_{\ell}}{\epsilon_{R}}}\). Using this result, we obtain \(\tilde{\mathbf{R}}^{\star}=\sqrt{\frac{\epsilon_{R}}{M_{\ell}}}\mathbf{I}_{M_ {\ell}}\). By multiplying (A3i) on the right-hand side by \((\tilde{\mathbf{Q}}+\tilde{\mathbf{Q}}^{\star})\) and using the condition \((\tilde{\mathbf{Q}}+\tilde{\mathbf{Q}}^{\star})\mathbf{\Gamma}_{Q}=\mathbf{0}\) in (A3k), it can be obtained that  
\[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]8 (A5)  
From the condition \(\mathbf{Z}^{-2}\mathbf{P}^{\star}(\tilde{\mathbf{Q}}+\tilde{\mathbf{Q}}^{\star })^{-1}=\mu_{P}\mathbf{P}^{\star}\) in (A3a), we can rewrite (A5) as  
\[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]9 (A6)  
Since the right-hand side of (A6) is Hermitian, the left-hand side of (A6) should be Hermitian. From this, we have \((\tilde{\mathbf{Q}}+\tilde{\mathbf{Q}}^{\star})\tilde{\mathbf{Q}}^{\star}= \tilde{\mathbf{Q}}^{\star}(\tilde{\mathbf{Q}}+\tilde{\mathbf{Q}}^{\star})\), or equivalently, \(\tilde{\mathbf{Q}}^{\star}\tilde{\mathbf{Q}}=\tilde{\mathbf{Q}}\tilde{ \mathbf{Q}}^{\star}\). Lemma A1: _[_41_]_ _For \(n\times n\) Hermitian matrices \(\mathbf{A}\) and \(\mathbf{B}\),_  
\[=J(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}},\mathbf{ \tilde{Q}})+\mu_{P}\Big{[}\text{Tr}(\mathbf{P}\mathbf{P}^{H})-P_{T}\Big{]}\]0  
_if and only if \(\mathbf{A}\) and \(\mathbf{B}\) share the same eigenvectors._  
From Lemma A1, we can conclude that \(\hat{\mathbf{Q}}\) and \(\tilde{\mathbf{Q}}^{\star}\) share the same eigenvectors. Let \(\hat{\mathbf{Q}}=\mathbf{U}_{Q}\mathbf{\Lambda}_{Q}\mathbf{U}_{Q}^{H}\) be the EVD of \(\hat{\mathbf{Q}}\). Then, \(\tilde{\mathbf{Q}}^{\star}\) should have the following form: \(\tilde{\mathbf{Q}}^{\star}=\mathbf{U}_{Q}\mathbf{D}_{Q}^{\star}\mathbf{U}_{Q} ^{H}\), where \(\mathbf{D}_{Q}^{\star}\) is an \(L\times L\) real diagonal matrix whose diagonal elements consist of the eigenvalues of \(\tilde{\mathbf{Q}}^{\star}\). From the structure of \(\tilde{\mathbf{Q}}^{\star}=\mathbf{U}_{Q}\mathbf{D}_{Q}^{\star}\mathbf{U}_{Q} ^{H}\), (A6) can be rewritten as  
\[=J(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}},\mathbf{ \tilde{Q}})+\mu_{P}\Big{[}\text{Tr}(\mathbf{P}\mathbf{P}^{H})-P_{T}\Big{]}\]1 (A7)  
Since the left-hand side of (A7) is diagonal, \(\mathbf{U}_{Q}^{H}\mathbf{P}^{\star H}\mathbf{P}^{\star}\mathbf{U}_{Q}\) should be diagonal.