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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?

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    $\begingroup$ Are some conditions missing? Unless all the singular values of $A$ are zeros or ones, $I-AA^T$ is not a projection. $\endgroup$ – user1551 Nov 21 '18 at 15:59
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    $\begingroup$ It is not a projection since $(I-AA^T)^2 \neq I-AA^T$. $\endgroup$ – LinAlg Nov 21 '18 at 16:01
  • $\begingroup$ 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. $\endgroup$ – Iamanon Nov 21 '18 at 16:09
  • $\begingroup$ 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. $\endgroup$ – Iamanon Nov 21 '18 at 16:17

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