# When is the nullspace unique in this case?

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 $$$$, then I believe this means the nullspace of $$I-AA^T$$ is $$$$?

Based on this information, is it possible to determine when $$$$ is unique?

• Are some conditions missing? Unless all the singular values of $A$ are zeros or ones, $I-AA^T$ is not a projection. – user1551 Nov 21 '18 at 15:59
• It is not a projection since $(I-AA^T)^2 \neq I-AA^T$. – LinAlg Nov 21 '18 at 16:01
• Hmmmm. I am initially not given any information about $A$. In a later problem, I see that $A$ has orthonormal columns and is not necessarily square. – Iamanon Nov 21 '18 at 16:09
• After consulting with a colleague, it seems that we are supposed to assume A has orthonormal columns. So it is a projection matrix based on this condition. – Iamanon Nov 21 '18 at 16:17