Question 15128
 I really cannot believe that anybody would ask questions about Jordan form,
 since I know that nodbody knows much about linear algebra here.

 {{{ Let T be a linear operator on V. 
Tv1= -v1-v2 and Tv2= v1-3v2 where (v1,v2) is a basis for V.
a) Find the characteristic and minimal polynomial.
b) Find all the linearly independent eigenvectors.
c) Find a Jordan basis for T and exhibit the Jordan form.  }}}
Sol: a) T(1, 0) = (-1,-1), T(0,1) = (1,-3)
(-v1 –v2) (-1  ,–1) (v1)
(v1 –3v2) (1  ,- 3) (v2)

 The matrix representation [T}B as (where B = {v1,v2} )
 (-1 ,1)
 (1, -3)

char(T) = det(xI –T) = (x+1)(x+3) + 1 = {{{x^2 +4x +4 =(x+2)^2 }}},
there is one double root: -2
So, the only eigenvalue of T is -2.
Since T + 2I = 
(1 ,-1)
(1, -1) is not the zero matrix.
And we know that the minimal polynomial min(T) is a factor of char(T) so, 
min(T) = char(T) = {{{(x+2)^2 }}}. 

b) To find the eigenvectors(eigenspace) for the eigenvalue -2,we have to solve Tx = -2 x for nonzero vectors x. 
That is, for the null space N(T+2I) of T+2I as:
Solve equations:
(1 –1) (x)   (0)
(1 –1) (y) = (0), we see that x - y = 0 .

 Hence, the eigenspace for the unique egenvalue–2 is <(1,1)>= {c(1,1) | c in R} (one dim. subspace generated by (1,1)). That is, the eigenspace has a basis  {(1,1)}

c) Since {(1,1)} is a basis of the N(T+2I) and T+2I is not the zero vector. 
Find a vector (x,y) such that (T + 2 I) (x, y) = (1,1). That is, x-y =1. Say, choose (1,0). Then, {(1,0), (1,1)} can form a Jordan basis for T and the corresponding Jordan form as 
(-2  0)
(1 –2).
[Check: T(1,0) = (-2,1) = -2(1,0) + (0,1) and T(1,1) = -2(0,1) correct.] 

 Try to read carefully about every details and study hard.
 I won't do too much for lazy students.

 Kenny
 PS.
 By the way, since min(T) = {{{(x+2)^2 }}}, we claim that {v, (T+2I)v} can be a Jordan basis for T if v is a vector such that (T+2I)v being non-zero.(i.e v is not in Null(T+2T)) (Why ?) 
This shows there are (infinitely) many Jordan basis for T. 
In fact, for this example, any v = (a,b) with a <> b , 
then {v, (1,1)} can be a Jordan basis for T.