Let $A$ $\in$ $R^{20\times20}$.
- rank($A$) = 18,
- rank($A^2$) = rank($A^3$) = 16,
- rank($A −E$) = 18,
- rank($(A − E)^2$) =16,
- rank($(A − E)^3$) = 14,
- rank($(A − E)^4$) = 12,
- rank($(A − E)^5$) = 11,
- rank($(A−E)^6$) = rank($(A−E)^7$) =10,
- rank($A+5E$) = 17,
- rank($(A+5E)^2$) = 15,
- rank($(A+5E)^3$) =rank($(A + 5E)^4$) = 14.
Find Jordan form of matrix $A$.
I don't really know what to do with this. I think the eigenvalues should be $\lambda=1$ and $\lambda=-5$, but after that I'm pretty much lost. I know how Jordan matrices work, I just don't know how to use them in this problem.
Hint: For any $\lambda \in \Bbb C$ and integer $k \geq 1$, $\operatorname{rank}((A - \lambda E)^k) - \operatorname{rank}((A - \lambda E)^{k-1})$ is the number of Jordan blocks within the Jordan form of $A$ that have size $k$ or larger.
Note: the sequence $\{\operatorname{rank}((A - \lambda E)^{k-1}) - \operatorname{rank}((A - \lambda E)^{k})\}_{k\geq 1}$ is sometimes called that "Weyr characteristic".