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Linear Algebra Homework Question

  1. Dec 12, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that A=[a,b;c,d]
    is diagonalizable if -4bc < (a-d)^2
    is not diagonalizale if -4bc > (a-d)^2

    2. Relevant equations

    For an nxn matrix, if there are n distinct eigenvalues then the matrix is diagonalizable.
    For an nxn matrix, if there are n linearly independent eigenvetors then the matrix is diagonalizable.

    3. The attempt at a solution

    Characteristic equation for eigenvalues:
    |λ-a,-b;-c,λ-d| = λ^2 + λ(a-d) + (ab - bc) = 0
    λ = 0.5 * (d - a plus-or-minus sqrt((a-d)^2 - 4ad + 4bc))

    For part 1, I need to show that there are two distinct solutions to this quadratic equation (b^2 - 4ac > 0: a,b,c from general quadratic eq - not from this problem)
    I know that (a-d)^2 + 4bc > 0

    However, I do not know how to show that (a-d)^2 - 4ad + 4bc > 0
     
    Last edited: Dec 12, 2011
  2. jcsd
  3. Dec 12, 2011 #2

    Mark44

    Staff: Mentor

    Expand the (a - d)2 term and then combine like terms. You're almost there!
     
  4. Dec 12, 2011 #3

    I like Serena

    User Avatar
    Homework Helper

    You've got a minus sign wrong in your characteristic equation.
     
  5. Dec 13, 2011 #4
    I had trouble with the same exact problem recently. Your process is correct. You just have a minus sign error.
     
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