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Jordan Normal Form

  1. Nov 18, 2006 #1
    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)?
     
    Last edited: Nov 18, 2006
  2. jcsd
  3. Nov 19, 2006 #2

    mathwonk

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    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.
     
  4. Nov 20, 2006 #3

    mathwonk

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    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?
     
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