Question about matrix with orthonormal columns.

Question

I met a question which says, if I have two matrix $A\in M_{n\times k}$ and $B\in M_{n\times l}$, both with orthonormal columns, with $n>k>l$. The question is, how to prove that $(AA^T - BB^T)x = x$ has a solution for some $x\neq0$?

Answer 1

Since the columns of $A$ are orthonormal, then $AA^T$ is the identity on any column on $A$. So if $V$ is the space generated by the columns of $A$, then $AA^T$ is the identity when restricted to $V$.

Let $W$ be the space generated by the columns of $B$. Since $k>l$, the dimension of $W^\perp$, wich is $n-l$ is bigger than $n-k$. So $V$ and $W^\perp$ have a nontrivial intesection.

Let $x\in V\cap W^\perp$. For such $x$ we have $AA^Tx=x$ and $BB^Tx=B(B^Tx)=B0=0$

So $(AA^T-BB^T)x=x$.