# Jordan Canonical Form

1. Oct 16, 2012

### Ressurection

1. The problem statement, all variables and given/known data
Let A be an nxn matrix (of real or complex components) and

$$J=\left(\begin{array}{c} λ & 1 & 0 & 0\\ 0 & λ & 1 & 0\\ & & ... & \\ 0 & 0 & λ & 1\\ 0 & 0 & 0 & λ &\end{array}\right) \,with\, λ \in ℂ$$

Show that there is
$$S = \left(\begin{array}{c} v1 & v2 & ... & v_n &\end{array}\right) \,with\, v1, v2, ...,v_n \inℂ$$
such that A = SJS-1 if and only if:
(A - λI)v1 = 0
(A - λI)vi+1 = vi , for i=1,2,....,n-1

2. Relevant equations

3. The attempt at a solution
My first step was A = SJS-1 ⇔ AS = SJ
Now, developing the right side I get SJ = [λv1 , v1 + λv2 ... vn-1 + λvn ]

So, column by column I get: Av1 = λv1 ⇔ (A-λ)v1 = 0
Av2 = v1 + λv2 ⇔ (A-λ)v2 = v1

and extending, I get
(A - λI)vi+1 = vi , for i=1,2,....,n-1

My only question is, does this solve the problem? I thought that to prove a ⇔ b , I had to prove a $\Rightarrow$ b and b $\Rightarrow$ a, but it seems to me that this proves both ways.