How to Find Bases for Jordan Canonical Forms

The idea for nding a basis relates to the proof of why a Jordan canonical form exists. What we seek to do is nd a largest possible set of chains (or cycles) of the form {x, (T −λkI)(x), . . . , (T −λkI)(x)} which are linearly independent. By the proof of Jordan canonical form, the number and lengths of these chains can be found from the numbers d0, . . . , d`k . Indeed, let λ = λk be a xed eigenvalue for T (α = αk, ` = `k) consider the transformation (T − λ)|Wλ on Wλ, the generalized eigenspace of T corresponding to λ. Then, let ei = dimR((T − λ)|iWλ) = dimWλ − di (by rank nullity) for 0 ≤ i ≤ `. Then, ei is a decreasing sequence with e0 = dimWλ, e` = 0, and e`−1 6= 0.