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Finding Invertible Matrix P

  1. Sep 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the Matrices A and B are similar

    [itex]A=\[ \left( \begin{array}{cc}
    1 & 1 \\
    -1 & 4 \end{array} \right)\] B=\[ \left( \begin{array}{cc}
    2 & 1 \\
    1 & 3 \end{array} \right)\] [/itex]



    2. Relevant equations

    B=PAP-1

    3. The attempt at a solution

    I know they have the same trace of 5

    I have found the charecteristic equations of both matrices which are the same

    [itex]\lambda^2-5\lambda+5[/itex]

    So I solve for zero to get the eigenvalues.

    [itex]\lambda^2-5\lambda+5=0[/itex] so the eigenvalues are

    [itex]\lambda_1=\frac{5+\sqrt{5}}{2}[/itex] and [itex]\lambda_2=\frac{5-\sqrt{5}}{2}[/itex]

    I'm having trouble finding the invertable Matrix P

    I know I have to use the matrix
    [itex]
    A=\[ \left( \begin{array}{cc}
    1-\lambda & 1 \\
    -1 & 4-\lambda \end{array} \right)\][/itex]

    and solve for zero. then I believe I use the diagonals of matrix B to substitute for [itex]\lambda[/itex] eg 2 and 3

    But not sure how to get there. Any help greatly appreciated
     
  2. jcsd
  3. Sep 28, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi boneill3! :smile:

    (have a lambda: λ and a square-root: √ and try using the X2 tag just above the Reply box :wink:)
    Do you need to find P, or even to find the eigenvalues?

    If the characteristic equations are the same, don't the matrices have to be similar?

    To find P, you could find the eigenvectors of each matrix first: then it's the matrix which transforms from one pair (as a basis) to the other pair.
     
  4. Sep 29, 2009 #3
    Hi Guy's
    I'm not sure if I have to find P but I would like to know how to do it.
    So I've found the eigenvectors of the two Matrices.

    The eigenvectors for
    [itex] A=\[ \left( \begin{array}{cc}1 & 1 \\-\frac{1}{2}(\sqrt{5}-3) & \frac{1}{2}(\sqrt{5}+3) \end{array} \right)\][/itex]

    [itex] B=\[ \left( \begin{array}{cc}1 & 1 \\-\frac{1}{2}(\sqrt{5}-1) & \frac{1}{2}(\sqrt{5}+1) \end{array} \right)\][/itex]


    I calculated D= P-1AP and D= P-1BP and both Matrices are diagonable.

    How do you find the Matrix which transforms one pair to the other?

    hence finding B=PAP-1

    I have not seen this yet in my course
    thanks
     
  5. Sep 29, 2009 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Do you mean you calculated Q and R such that D= Q-1AQ and D= R-1BR with D diagonal?

    Then B = RDR-1 = RQ-1AQR-1 = (QR-1)-1A(QR-1) :wink:
     
  6. Sep 29, 2009 #5
    Thanks Tiny_Tim, Your a Ledgend!

    I found the change of basis Matrix P to be

    [itex]P=\[ \left( \begin{array}{cc}1 & 1 \\1 & 0 \end{array} \right)\][/itex]

    B = PAP-1

    Using form what you said that P = QR-1

    It makes heaps more sense to me know. Yippieeeee!!!
     
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