Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form $$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\cdots &\bf{0}\\ \bf{0}&{\bf{w}}_2 & \bf{0}& \cdots &\cdots \\ \vdots &\vdots&\vdots&\vdots &\vdots\\ \bf{0}& \cdots &\cdots & \bf{0} &{\bf{w}}_m \end{array} \right]$$ where all ${\bf{w}}_n$ are $1\times \ell$ nonzero vectors, and $\bf{A}$ an $m\ell\times t$ matrix that I generate randomly according to an entrywise IID complex Gaussian distribution.
Set $\bf{Z}=\bf{QWA}$. Assume $k>\ell$ and $m>t$. For fixed values of $m, k, \ell$, I need to show that it is always possible to select $\bf{W}$ such that ${\mathrm{rank}}({\bf{Z}}) \leq t-k$ whenever $\ell\geq \ell_{\min}$.
To carry on my work, I need to characterize $\ell_{\min}$ in terms of $m, k, \ell$, but I am stuck....from numerical search, I can compute such $\bf{W}$, but I fail to derive the exact relation between $\ell_{\min}$ and $m, k, \ell$.
