Any singular matrix $A$ is similar to a matrix of form $\begin{pmatrix}B&\\&0\end{pmatrix}$, where $B$ is a matrix with rank $r$?
Is this right? that is, is there exists an invertible matrix $P$, such that $P^{-1}AP=\begin{pmatrix}B&\\&0\end{pmatrix}$, with $rank B=r$.
$B$ may not be a $r\times r$ matrix? Or may we assume $B$ is $r\times r$?
As is well known, invertible $P,Q$ exists such that $PAQ=\begin{pmatrix}I_r&\\&0\end{pmatrix}$. Does this help?