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Find a Matrix P that diagonalizes A

  1. Apr 4, 2014 #1
    1. The problem statement, all variables and given/known data
    Find a Matrix P that diagonalizes A


    2. Relevant equations

    A = [tex]
    \begin{pmatrix}
    2 & 0 & -2\\
    0 & 3 & 0\\
    0 & 0 & 3
    \end{pmatrix}
    [/tex]

    3. The attempt at a solution


    Well right off the bat we know that this is an upper triangular matrix so the eigenvalues are the entries along the main diagonal of A.

    So λ = 2, 3, 3

    But if an n x n matrix A has n distinct eigenvalues, then A is diagonalizable.

    In this case we only have 2 distinct eigenvalue so it shouldnt be diagonalizable.....

    But the answer is:
    P = [tex]
    \begin{pmatrix}
    -2 & 0 & 1\\
    0 & 1 & 0\\
    1 & 0 & 0
    \end{pmatrix}
    [/tex]


    What is the proper way to start the problem to find this matrix P??????

    Thank You for any help in advance.
     
  2. jcsd
  3. Apr 5, 2014 #2

    Mark44

    Staff: Mentor

    If you check, you'll see that the eigenspace of λ = 3 is of dimension 2, so there are two eigenvectors for this eigenvalue.

    When a matrix has repeated eigenvalues, there is some terminology that distinguishes between algebraic multiplicity vs. geometric multiplicity. I think this matrix is a case of geometric multiplicity.

    To find your matrix P, the important thing is to get three linearly independent eigenvectors, not whether there are three distinct eigenvalues.
     
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