We know two matrices $A \in \mathcal{R}^{n \times n}$ and $B \in \mathcal{R}^{n \times n}$ are said to be similar, iff there exists a matrix $P$ such that $A= PB P^{-1}$.
Can this be extended to include pseudoinverses of rectangular matrices. Thus, given two square matrices of different sizes $C \in \mathcal{R}^{n \times n}$ and $D \in \mathcal{R}^{m \times m}$, and a transformation matrix $Q \in \mathcal{R^{n \times m}}$, can $C=QD Q^+$ be defined as a legitimate similarity transformation?
Is there a name for this?