Calculating JNF & Basis Vectors: Jordan Normal Form

[pivoxa15]

1. If given a matrix in JNF, what would be its basis? How would you calculate it?
If you put the basis vectors (of JNF) as columns in matrix P than
P=(T^-1)PA, T and A are given.

where T is the original matrix and A is T in JNF. But I cannot explicitly calculate P since it is on both sides. How do I find P?

2. If the minimal polynomial is given as (x+2)(x-4)=0, and the -2 eigenvalue results in a 0 dimensional null space (i.e. (0,0,0) vector) what would the JNF look like given the null space of eigenvalue 4 is 2 dimensional. And the original matrix is 3 by 3.

Would it be
diag(0,4,4)?

 

[mathwonk]

the question does not quite make sensfinding P is a bit of work and hard to say briefly.

i can say it briefly but it won't help you that much.

 

[mathwonk]

briefly, to find a jordan basis for a map T with minimalpolynomial (X-a)^r, find a basis for ker(T-a)^r/ker(T-a)^(r-1), extend to basis of ker(T-a)^(r-1)/ker(T-a)^(r-2),...

see what I mean?