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Jordan forms of a matrix

  1. Nov 29, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose that the matrix A has characteristic equation (lambda - 2)^3 * (lambda + 1)^2

    (a) Write all 6 of the possible Jordan forms of A.
    (b) Compute det(A) and tr(A).


    2. Relevant equations



    3. The attempt at a solution
    To figure out Jordan forms I need to find the eigenvalues and the algebraic and geometric multiplicities, right? The eigenvalue of 2 has algebraic multiplicity of 3, and the eigenvalue -1 has algebraic multiplicity of 2. But since I don't know the form of the matrix how do I figure out the geometric multiplicities?

    thanks
     
  2. jcsd
  3. Nov 29, 2008 #2
    What are the possible elementary divisors of A? You should get six possibilities and each possibility will yield a particular Jordan form.
     
  4. Nov 30, 2008 #3
    I looked through some examples from notes, and it seems like the geometric multiplicity depends on the algebraic. i.e. if alg multiplicty = 2, then geometric multiplicty = 1 or 2. if alg multiplicity = 3, then geometric multiplicity = 1, 2, or 3. I'm not sure if this is right, but this happens in all the examples I looked at. But this only leaves me with 5 possible Jordan forms?
     
  5. Nov 30, 2008 #4
    The alg. mult. is always >= the geom. mult. How are you getting 5? Are you just counting the possible geom. mult. of given alg. mult. of 2 and 3?

    Forget about the multiplicities for a moment. What are the eigenvalues of A?
     
  6. Nov 30, 2008 #5
    2 and -1?
     
  7. Nov 30, 2008 #6
    Right. And what are the algebraic multiplicities of each? What are the possible geometric multiplicities of each? What are the possible geometric multiplicities of 2 and -1 together?
     
  8. Dec 2, 2008 #7
    i think you can potentially have the jordan forms [2 0 0 0 0; 0 2 0 0 0; 0 0 2 0 0; 0 0 0 -1 0; 0 0 0 0 -1] ;;; [2 1 0 0 0; 0 2 0 0 0; 0 0 2 0 0; 0 0 0 -1 0; 0 0 0 0 -1] ;;; [2 1 0 0 0; 0 2 1 0 0; 0 0 2 0 0; 0 0 0 -1 0; 0 0 0 0 -1] ;;; [2 0 0 0 0; 0 2 1 0 0; 0 0 2 0 0; 0 0 0 -1 0; 0 0 0 0 -1] ;;; [2 0 0 0 0; 0 2 0 0 0; 0 0 2 0 0; 0 0 0 -1 1; 0 0 0 0 -1] ;;; [2 1 0 0 0; 0 2 1 0 0; 0 0 2 0 0; 0 0 0 -1 1; 0 0 0 0 -1]

    can someone confirm this? btw, are you in berg's class?
     
  9. Dec 2, 2008 #8
    The fourth Jordan form you listed is the same as the second: [2 0 0 0 0; 0 2 1 0 0; 0 0 2 0 0; 0 0 0 -1 0; 0 0 0 0 -1].
     
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