1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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)


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


    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


    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook