I am a little confused about the following:
Let A be an $\mathbb{R}^{n \times f}$ matrix, B be an $\mathbb{R}^{m \times f}$ matrix, and C be an $\mathbb{R}^{f \times d}$ orthogonal matrix. If $f > d$, then $A B^T \neq (A C) (B C)^T$. However, whenever $d \leq f$, the statement becomes true, i.e. $A B^T = (A C) (B C)^T$.
I probably miss a very straightforward property here since I have still a lot to learn in linear algebra. I would appreciate it if you can help me figure out the underlying reason here.
Thanks.