I've been staring at a chapter in Bill Cooperstein's Advanced Linear ALgebra for some time now and one section is giving me trouble. It is about elementary divisors and invariant factors. My question is this:
Let $S$ be an operator on a finite dimensional real vector space and assume that $U=[S,\mathbf{u}_1] \oplus [S,\mathbf{u}_2] \oplus\cdots\oplus [S,\mathbf{u}_6]$ where $[S,\mathbf{u}_i]$ is the $S$-cyclic subspace generated by $\mathbf{u}_i$, where $\mu_{S,\mathbf{u}_i}(x)$ is the minimal polynomial of each $S$-cyclic subspace
and \begin{align*} \mu_{S,\mathbf{u}_1}(x) &= \mu_{S,\mathbf{u}_2}(x) = (x^2+1)^5,\\ \mu_{S,\mathbf{u}_3}(x) &= (x^2+1)^4,\\ \mu_{S,\mathbf{u}_4}(x) &= \mu_{S,\mathbf{u}_5}(x) = (x^2+1)^2,\\ \mu_{S,\mathbf{u}_6}(x) &= x^2+1. \end{align*} Set $U_i = \{\mathbf{u}\in U \mid (S^2+I_U)^i(\mathbf{u}) = 0\}$ for $i = 1,\ldots,6$.
Determine the dimension of each $U_i$.
We know each minimal polynomial is irreducible but they are not distinct from each other. What does this tell us about each $S$-cyclic subspace for vector $\mathbf{u}_i$?