# Minimum rank of a product of two block diagonal matrices with an arbitrary matrix

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 $$$$\boldsymbol{W}= \begin{bmatrix} \boldsymbol{W}_1 & \boldsymbol{0}_{m_1\times l} & \dots & \boldsymbol{0}_{m_1\times l}\\ \boldsymbol{0}_{m_2\times l} & \boldsymbol{W}_2 & \dots & \boldsymbol{0}_{m_2\times l}\\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{0}_{m_b\times l} & \boldsymbol{0}_{m_b\times l} & \dots & \boldsymbol{W}_b \end{bmatrix},$$$$ where where $$\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

$$$$\boldsymbol{Q}= \begin{bmatrix} \boldsymbol{Q}_1 & \boldsymbol{0}_{l\times n_1} & \dots & \boldsymbol{0}_{l\times n_1}\\ \boldsymbol{0}_{l\times n_2} & \boldsymbol{Q}_2 & \dots & \boldsymbol{0}_{l\times n_2}\\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{0}_{l\times n_p} & \boldsymbol{0}_{l\times n_p} & \dots & \boldsymbol{Q}_p \end{bmatrix},$$$$ where where $$\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 $$\boldsymbol{WHQ}$$?