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How to find invertible matrix and diagonal matrix

  1. Oct 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Find an invertible matrix P and a diagonal matrix D such that D=P^(-1)AP.

    A=
    (−1 2 −2)
    (−2 3 −2)
    ( 2 −2 3)

    2. Relevant equations


    3. The attempt at a solution

    the eigenvalues are 1, 1, and 3
    the eigenvector i've found so far is for the eigenvalue 3, which is t(1, -1, 1)
    when i was attempting to find the eigenvector of 1,

    i did (1*I-A)(x), and then reduced the matrix to

    (1 1 -1)
    (0 0 0)
    (0 0 0)
    and in polynomial form it would be
    x+y-z=0

    how am i suppose to put this in homogeneous form? x=(a, b, c)t

    thanks in advance!!
     
  2. jcsd
  3. Oct 16, 2009 #2

    Mark44

    Staff: Mentor

    Your equation x + y + z = 0 will give you two solutions that correspond to the two eigenvectors you need for this eigenvalue.

    x = - y - z
    y = y
    z = .......z

    So (x, y, z) = y(-1, 1, 0) + z(-1, 0, 1)

    y and z are parameters. For one eigenvector, choose y = 1, z = 0. For the other, choose y = 0, z = 1.
     
  4. Oct 16, 2009 #3
    yeah i tried that,
    our assignment is online based and it tells you immediately if its right or wrong..
    however ive tried everything but nothing works!
    does the order for the eigenvalues matter? (x1, x2,and x3)
    does the engevectors have to be alligned in a specific way for A=PDP^-1 to be valid?
     
  5. Oct 16, 2009 #4

    Mark44

    Staff: Mentor

    The order of the columns in your matrix P (and hence P-1) determine where the eigenvalues appear where in your diagonal matrix. Is that what you're asking?
     
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