Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times l \cdot b$ block diagonal matrix, $\boldsymbol{W}$, of the form \begin{equation} \boldsymbol{W}= \begin{bmatrix} \boldsymbol{W}_1 & \boldsymbol{0}_{m_1\times l} & \dots & \boldsymbol{0}_{m_1\times l}\\ \boldsymbol{0}_{m_2\times l} & \boldsymbol{W}_2 & \dots & \boldsymbol{0}_{m_2\times l}\\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{0}_{m_b\times l} & \boldsymbol{0}_{m_b\times l} & \dots & \boldsymbol{W}_b \end{bmatrix}, \end{equation} where where $\boldsymbol{W}_i$ are $m_i \times l$ matrices with $\sum_{i=1}^bm_i=m$, $1\leq m_i < l$, and $\boldsymbol{0}_{m_i\times l}$ denotes the $m_i\times l$ all-zero matrix. We also have an $l \cdot p\times n$ block diagonal matrix, $\boldsymbol{Q}$, of the form
\begin{equation} \boldsymbol{Q}= \begin{bmatrix} \boldsymbol{Q}_1 & \boldsymbol{0}_{l\times n_1} & \dots & \boldsymbol{0}_{l\times n_1}\\ \boldsymbol{0}_{l\times n_2} & \boldsymbol{Q}_2 & \dots & \boldsymbol{0}_{l\times n_2}\\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{0}_{l\times n_p} & \boldsymbol{0}_{l\times n_p} & \dots & \boldsymbol{Q}_p \end{bmatrix}, \end{equation} where where $\boldsymbol{Q}_i$ are $l \times n_i$ matrices with $\sum_{i=1}^pn_i=n$, $1\leq n_i < l$. Both $\boldsymbol{W}$ and $\boldsymbol{Q}$ can be selected for each possible $\boldsymbol{H}$, but they should be selected in such a way that they are both full rank, i.e., $\mathrm{rank}(\boldsymbol{W})=m$ and $\mathrm{rank}(\boldsymbol{Q})=n$.
What is the minimum rank of the matrix product $\boldsymbol{WHQ}$?
