
Direct Sums
"Suppose that T : V > V is a linear transformation of vector spaces over
R whose minimal polynomial has no multiple roots. Show that V can be
expressed as a direct sum
V = V1 + V2 + · · · + Vt
of Tstable subspaces of dimensions at most 2. Show that, relative to a suitable basis, T can be represented by an n × n matrix with at most 2n nonzero entries, where n := dim(V)."
If only there was a way to represent the complex roots of the minimal polynomial with 2x2 matrices all call the whole 2x2 matrix an "eigenvalue". If not, I don't know what to do.

Re: Direct Sums
Think about this.
If the minimal polynomial is $\displaystyle m(x)=\prod_j (xz_j)$, we can define $\displaystyle V_j^1=\{v: Tv=Re(z_j) v\}, V_j^2=\{v: Tv=Im(z_j) v\}$, and $\displaystyle V_j=V_j^1+V_j^2$.