# RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
## Appendix A  
To solve (8) numerically via the Lagrangian approach, we consider the augmented Lagrangian:  
\[L_{1}(G,\widetilde{G}) :=\tfrac{1}{2}\|\mathcal{R}(\operatorname{Re}(F_{1})G-I)\|_{F}^{2}\] \[\quad+\operatorname{Tr}\left(\lambda_{1}^{\top}(\widetilde{G}G- I)\right)+\operatorname{Tr}\left(\lambda_{2}^{\top}(G\widetilde{G}-I)\right)\] \[\quad+\tfrac{\rho}{2}\Big{(}\|\widetilde{G}G-I\|_{F}^{2}+\|G \widetilde{G}-I\|_{F}^{2}\Big{)},\]  
with fixed \(\rho>0\). We initialize \(G_{R}\) as in (4), \(\widetilde{G}\) as \(G_{R}^{-1}\), and \(\lambda_{1}=\lambda_{2}\) as zero matrices, and iterate the updates until convergence of \(\lambda_{1}\) and \(\lambda_{2}\):  
(i) \(G_{R}\leftarrow\operatorname{argmin}_{G}L_{1}(G,\widetilde{G})\);(ii) \(\widetilde{G}\leftarrow\operatorname*{argmin}_{\widetilde{G}}L_{1}(G_{R}, \widetilde{G})\); (iii) \(\lambda_{1}\leftarrow\lambda_{1}+\rho(\widetilde{G}G_{R}-I)\) and \(\lambda_{2}\leftarrow\lambda_{2}+\rho(G_{R}\widetilde{G}-I)\). where steps (i) and (ii) can be approximated by solving the Sylvester equations \(\
abla_{G}L_{1}=0\) and \(\
abla_{\widetilde{G}}L_{1}=0\) numerically (e.g., via matlab function sylvester), where:  
\[\
abla_{G}L_{1} =\operatorname{Re}(F_{1})^{\top}\mathcal{R}(\operatorname{Re}(F_{ 1})G-I)+\widetilde{G}^{\top}\lambda_{1}+\lambda_{2}\widetilde{G}^{\top}\] \[\quad+\rho(\widetilde{G}^{\top}(\widetilde{G}G-I)+(G\widetilde{G }-I)\widetilde{G}^{\top})\] \[=(\operatorname{Re}(F_{1})^{\top}\mathcal{R}\operatorname{Re}(F_ {1})+\rho\widetilde{G}^{\top}\widetilde{G})G+G(\rho\widetilde{G}\widetilde{G} ^{\top})\] \[\quad+(\widetilde{G}^{\top}\lambda_{1}+\lambda_{2}\widetilde{G} ^{\top}-\operatorname{Re}(F_{1})^{\top}\mathcal{R}-2\rho\widetilde{G}^{\top}),\]  
and  
\[\
abla_{\widetilde{G}}L_{1} =G^{\top}\lambda_{2}+\lambda_{1}G^{\top}+\rho((\widetilde{G}G-I )G^{\top}+G^{\top}(G\widetilde{G}-I))\] \[=\rho(G^{\top}G\widetilde{G}+\widetilde{G}GG^{\top})+G^{\top} \lambda_{2}+\lambda_{1}G^{\top}-2\rho G^{\top}.\]  
Problem (9) is solved analogously, now considering:  
\[L_{2}(H,\widetilde{H}):= \tfrac{1}{2}\|D_{1}-G_{R}AH\|_{F}^{2}+\tfrac{v}{2}\|H-\widetilde{ H}^{\top}\|_{F}^{2}\] \[\quad+\operatorname{Tr}\Bigl{(}\lambda_{3}^{\top}(\widetilde{H} H-I)\Bigr{)}+\operatorname{Tr}\Bigl{(}\lambda_{4}^{\top}(H\widetilde{H}-I) \Bigr{)}\] \[\quad+\tfrac{\mu}{2}\Bigl{(}\|\widetilde{H}H-I\|_{F}^{2}+\|H \widetilde{H}-I\|_{F}^{2}\Bigr{)},\]  
initializing \(H_{R}\) as in (4), \(\widetilde{H}\) as \(H_{R}^{\top}\) and \(\lambda_{3}=\lambda_{4}\) as zero matrices. The iteration steps are now:  
(i') \(H_{R}\leftarrow\operatorname*{argmin}_{H}L_{2}(H,\widetilde{H})\); (ii') \(\widetilde{H}\leftarrow\operatorname*{argmin}_{\widetilde{H}}L_{2}(H_{R}, \widetilde{H})\); (iii') \(\lambda_{3}\leftarrow\lambda_{3}+\mu(\widetilde{H}H_{R}-I)\) and \(\lambda_{4}\leftarrow\lambda_{4}+\mu(H_{R}\widetilde{H}-I)\), and the derivatives concerning (i') and (ii') are:  
\[\
abla_{H}L_{2} =A^{\top}G^{\top}(D-GAH)+v\widetilde{H}(H-\widetilde{H}^{\top})\] \[\quad+\widetilde{H}^{\top}\lambda_{3}+\lambda_{4}\widetilde{H}^{\top}\] \[\quad+\mu(\widetilde{H}^{\top}(\widetilde{H}H-I))+\mu((H\widetilde {H}-I)\widetilde{H}^{\top})\] \[=(A^{\top}G^{\top}GA+v\widetilde{H}+\mu\widetilde{H}^{\top} \widetilde{H})H+H(\mu\widetilde{H}\widetilde{H}^{\top})\] \[\quad+A^{\top}G^{\top}D+\widetilde{H}^{\top}\lambda_{3}+\lambda_{ 4}\widetilde{H}^{\top}-2\mu\widetilde{H}^{\top}-v\widetilde{H}\widetilde{H} ^{\top},\]  
and  
\[\
abla_{\widetilde{H}}L_{2} =vH(\widetilde{H}-H^{\top})+\lambda_{3}H^{\top}+H^{\top}\lambda_{4}\] \[\quad+\mu((\widetilde{H}H-I)H^{\top})+\mu(H^{\top}(H\widetilde{H} -I))\] \[=(vH+\mu H^{\top}H)\widetilde{H}+\widetilde{H}(\mu HH^{\top})\] \[\quad+(\lambda_{3}H^{\top}+H^{\top}\lambda_{4}-2\mu H^{\top}-vHH^{ \top}).\]