# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Mathematical Preliminaries
: _Having the equality \(\mathbf{C}=\mathbf{H}_{r}\mathbf{Q}_{r}\mathbf{H}_{r}^{H}\), one can generally conclude the rule of multiplication is determinant, i.e., \(\det(\mathbf{C})=\det(\mathbf{H}_{r}^{H}\tilde{\mathbf{H}}_{r})\det(\mathbf{ Q}_{r})\). Further, using the properties of determinants we can also conclude \(\prod_{i=1}^{N}\lambda_{i}(\mathbf{C})=\prod_{i=1}^{N}\left(\lambda_{\rho(i)}( \mathbf{H}_{r}^{H}\mathbf{H}_{r})\lambda_{i}(\mathbf{Q}_{r})\right)\) where \(\rho(i)\) is a random permutation of \(i\) and indicates that there is no need for \(\lambda_{\rho(i)}\left(\mathbf{H}_{r}^{H}\tilde{\mathbf{H}}_{r}\right)\) to be in decreasing order. However, one cannot generally conclude \(\lambda_{i}(\mathbf{C})=\lambda_{\rho(i)}(\mathbf{H}_{r}^{H}\tilde{\mathbf{H}}_{r}) \lambda_{i}(\mathbf{Q}_{r}),\forall i\), unless \(\mathbf{H}_{r}^{H}\mathbf{H}_{r}\) and \(\mathbf{Q}_{r}\) share common basis._  
As a result of Remark 1, in general, we cannot rewrite (26) in terms of \(\lambda_{i}\!\left(\mathbf{H}_{r}^{H}\mathbf{H}_{r}\right)\), \(\lambda_{i}\!\left(\mathbf{Q}_{r}\right)\), \(\lambda_{i}\!\left(\mathbf{H}_{r}^{H}\mathbf{H}_{1}\right)\) and \(\lambda_{i}\!\left(\mathbf{Q}_{s}\right)\). However, if we show that for every choice of \(\mathbf{Q}_{s}\), there exists a matrix \(\mathbf{Q}_{s}^{\prime}\) with properties: (1) \(\lambda_{i}\!\left(\mathbf{H}_{1}\mathbf{Q}_{s}\mathbf{H}_{1}^{H}\right)=\lambda_ {i}\!\left(\mathbf{H}_{1}\mathbf{Q}_{s}^{\prime}\mathbf{H}_{1}^{H}\right)\); (2) \(\lambda_{i}\!\left(\mathbf{H}_{1}\mathbf{Q}_{s}^{\prime}\mathbf{H}_{1}^{H}\right)= \lambda_{i}\!\left(\mathbf{Q}_{s}^{\prime}\right)\lambda_{i}\!\left(\mathbf{H}_ {1}^{H}\mathbf{H}_{1}\right)\) and (3) \(\text{Tr}\!\left(\mathbf{Q}_{s}^{\prime}\right)\leq\text{Tr}\!\left(\mathbf{Q}_{s }\right)\); then we can use \(\mathbf{Q}_{s}^{\prime}\) instead and rewrite (26) in terms of \(\lambda_{i}\!\left(\mathbf{H}_{1}^{H}\mathbf{H}_{1}\right)\) and \(\lambda_{i}\!\left(\mathbf{Q}_{s}^{\prime}\right)\) to simplify the problem. The first property implies that both \(\mathbf{Q}_{s}\) and \(\mathbf{Q}_{s}^{\prime}\) have the exact same impact on the capacity. Hence, if we find a \(\mathbf{property means, unlike \(\mathbf{Q}_{s}\), \(\mathbf{Q}_{s}^{\prime}\) actually shares the common basis with \(\mathbf{H}_{1}^{H}\mathbf{H}_{1}\). The last property implies that \(\mathbf{Q}_{s}^{\prime}\) is at least as good as \(\mathbf{Q}_{s}\) in terms of power consumption. Observe that if we show for every feasible \(\mathbf{Q}_{s}\) there exists at least one such \(\mathbf{Q}_{s}^{\prime}\), then we can solve the problem (14) in a much easier way. The reason is, in such a case, instead of searching for optimal \(\mathbf{Q}_{s}\) over the whole feasible set, we can search for the optimal \(\mathbf{Q}_{s}^{\prime}\). Unlike \(\mathbf{Q}_{s}\), finding \(\mathbf{Q}_{s}^{\prime}\) does not need a complete search over the whole feasible set since \(\mathbf{Q}_{s}^{\prime}\) shares a common basis with \(\mathbf{H}_{1}^{H}\mathbf{H}_{1}\). Therefore, we can limit our search only to the portion of the feasible set in which the matrices have eigendirections identical to those of \(\mathbf{H}_{1}^{H}\mathbf{H}_{1}\). Similarly, if we show for every choice of \(\mathbf{H}_{r}\), there exist at least one \(\mathbf{H}_{r}^{\prime}\) for which we have three conditions \(\lambda_{i}(\mathbf{H},\mathbf{Q}_{r}\mathbf{H}_{r}^{H})=\lambda_{i}(\mathbf{ H}_{r}^{\prime}\mathbf{Q}_{r}\mathbf{H}_{r}^{H})\), \(\lambda_{i}(\mathbf{H},\mathbf{Q}_{r}\mathbf{H}_{r}^{H})=\lambda_{i}(\mathbf{ Q}_{r})\lambda_{i}(\mathbf{H}_{1}^{H}\mathbf{H}_{1}^{H})\) and \(\text{Tr}(\mathbf{H}_{1}^{H}\mathbf{H}_{1}^{H})\leq\text{Tr}(\mathbf{H}_{1}^{H }\mathbf{H}_{1})\), we can simplify our search to finding \(\mathbf{H}_{r}^{\prime}\) instead of \(\mathbf{H}_{r}\). In the next theorem, we show that such \(\mathbf{Q}_{s}^{\prime}\) and \(\mathbf{H}_{r}^{\prime}\) exist.