I am given the following: $$ I - AA^T $$ is a projection matrix onto the orthogonal complement of $< A >$.
So the nullspace of $I-AA^T$ is the subspace spanned by the set of vectors $x$ such that $(I-AA^T)x = 0$. Since we know the projection matrix projects x onto the orthogonal complement of $<A>$, then I believe this means the nullspace of $I-AA^T$ is $<A>$?
Based on this information, is it possible to determine when $<A>$ is unique?