Determine the rank of $A$ if $A$ is a positive $n\times n$ matrix with $A^2 = A$. Give a geometric interpretation of $A$. What happens in the case when $A^k = A$ for some integer $k\geq 3$?
Clarifying, positive matrix ($A > 0$ or $A \succ 0$, since it might be confused with positive definiteness) is a matrix in which each element is positive.
I know from previous exercises that $\operatorname{rank} A = \operatorname{trace}A^{k-1}$ if $A^k = A$, but that holds without $A$ being positive so I assume that there is more to say about the rank. And likewise, I know that $A^2=A$ defines a projections, but that holds without $A$ being positive. So my question is how should one start, alternatively continue with said properties?
Perron's Theorem is known, but there is not more to work with.
Hint: If $A^2 = A$, then $A$ is diagonalizable with eigenvalues $0$ and $1$. However, by the Perron theorem, the multiplicity of the eigenvalue $1$ is just $1$.
Similarly, if $A^k = A$, note By the Perron theorem that $A$ has the eigenvalue $1$ with multiplicity $1$ and no other eigenvalues of magnitude $1$.