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Linear algebra: find a matrix P that satisfies D=P(inv)AP with known matrices A and D

  1. Mar 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Find a matrix P that satisfies D=P[itex]^{-1}[/itex]AP (A and D are similar)

    A=

    2 2 -1
    1 3 -1
    -1 -2 2

    D=

    1 0 0
    0 1 0
    0 0 5

    2. Relevant equations



    3. The attempt at a solution

    OK, so I know how to find a matrix P for A, but I DONT know how to find the specific P that gets the specific A.. anyways here is my work so far

    Since A and D are similar, e-values of D are the same as those of A

    It is easy to find the e-values of D --> det([itex]\lambda[/itex]I-D)=0
    so ([itex]\lambda[/itex]-1)([itex]\lambda[/itex]-1)([itex]\lambda[/itex]-5)=0
    so e-values are 1, 1, 5

    So I found e-vector of A using [itex]\lambda[/itex]=1

    I did this by solving for vector x in: ([itex]\lambda[/itex]I-A)x=0

    I found the following vector: x=t[-2, 1, 0]+w[1, 0, 1] where t and w are elements of the reals

    doing the same for [itex]\lambda[/itex]=5 I get: x=t[-1 -1 1]

    so a P for A (not necessarily the proper P) is

    -2 1 -1
    1 0 -1
    0 1 1

    Using the same procedure for D as for A above, I get a P for D to be

    0 0 0
    0 1 0
    1 0 1

    This is where I have no idea what to do. I remember vaguely reading somewhere that the P in question is the matrix that transforms the P for A from above to the P for D

    so I solve

    P[itex]_{A}[/itex]P=P[itex]_{D}[/itex] and get the following as a matrix

    0.25 0.50 0.25
    0.75 0.50 0.75
    0.25 -0.5 0.25

    ...but checking in D=P[itex]^{-1}[/itex]AP, the matrix P above isn't even invertible. Where did I go wrong? Thanks!
     
  2. jcsd
  3. Mar 24, 2012 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Linear algebra: find a matrix P that satisfies D=P(inv)AP with known matrices A a

    Try applying this to A. The matrix P is known as the matrix that diagonalizes A. There isn't much point in trying to find a different matrix to diagonalize D.
     
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