Conditions on a matrix having 1 as eigenvalue

Question

Suppose $A\in\mathbb{R}^{m \times n}$, $m\geq n$, and assume $A (A^TA)^{-1}A^T$ has 1 as an eigenvalue. Is it possible to say anything about the structure of A?

Answer 1

$P := A(A^\top A)^{-1} A^\top$ is the orthogonal projection onto the column space of $A$. Orthogonal projections are idempotent ($P^2=P$) and symmetric ($P^\top = P$) and you can check that the only eigenvalues of $P$ are $0$ and $1$. So actually, $1$ is always an eigenvalue of your matrix (except in the case when $A$ is the zero matrix).