Setup: An $(\omega,\Lambda)$ base matrix $W\in\mathbb{R}^{R\times C}$ is described by the coupling width $\omega\geq1$ and the coupling length $\Lambda\geq2\omega-1$. The matrix has $R=\Lambda+\omega-1$ rows and $C=\Lambda$ columns, with the $(r,c)$-th entry of the matrix, where $r\in\{1,\dots,R\}$ and $c\in\{1,\dots,C\}$, given by $$ W_{rc}= \begin{cases} 1/\omega &\text{if $c\leq r\leq c+\omega-1$,} \\ 0 &\text{otherwise.} \end{cases} $$ It is known that we can create a Gaussian spatially coupled matrix $A\in\mathbb{R}^{n\times p}$ (visually, a band diagonal matrix), in the following way:
- First specify a base matrix $W$ of dimensions $R\times C$.
- Replace each entry of the base matrix $W_{rc}$ with an $\frac{n}{R}\times\frac{p}{C}$ matrix with entries drawn i.i.d. $\sim N\big(0,\frac{W_{rc}}{n/R}\big)$.
I would like to extend this idea to rotationally invariant matrices $A$ where $$ A=O^\top\Gamma Q \in\mathbb{R}^{n\times p}, $$ where $\Gamma=\text{diag}(\lambda)$ where $\lambda$ are the singular values, and $O$ and $Q$ are independent Haar orthogonal matrices.
Motivation: This would allow us to design more general codes for communication etc.
Question: Can we use the same approach of replacing entries of the base matrix $W$ to produce a band diagonal (spatially coupled) matrix $A$ where each sub-matrix of non-zero entries $\widetilde{A}$ in $A$ can be written in the form of $\widetilde{A}=\widetilde{O}^\top\widetilde{\Gamma}\widetilde{Q}$ (where $\widetilde{O}$, $\widetilde{\Gamma}$, and $\widetilde{Q}$ are some matrices)? Refer to the image below for a visual representation of my question. If so, then how can I do it? Thanks.
