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Math Help - diagonal matrix

  1. #1
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    diagonal matrix

    Find an invertible matrix P and a diagonal matrix D so that (P^−1)AP = D, where
    A =
    4 2
    3 3
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  2. #2
    Eater of Worlds
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    I'll start you off.

    \Large\begin{bmatrix}4&2\\3&3\end{bmatrix}

    Find your charpoly:

    det\left({\lambda}\begin{bmatrix}1&0\\0&1\end{bmat  rix}-\begin{bmatrix}4&2\\3&3\end{bmatrix}\right)=\begin  {bmatrix}{\lambda}-4&-2\\-3&{\lambda}-3\end{bmatrix}={\lambda}^{2}-7{\lambda}+6

    The roots of this quadratic are your eigenvalues.

    They are 1 and 6.

    Sub them in for lambda in the matrix we derived, getting:

    \begin{bmatrix}-3&-2\\-3&-2\end{bmatrix}\;\text{and}\;\begin{bmatrix}2&-2\\-3&3\end{bmatrix}

    Use rref on these matrices and we find:

    \begin{bmatrix}-3&-2\\-3&-2\end{bmatrix}=\begin{bmatrix}1&\frac{2}{3}\\0&0\e  nd{bmatrix}\;\text{and}\;\begin{bmatrix}2&-2\\-3&3\end{bmatrix}=\begin{bmatrix}1&-1\\0&0\end{bmatrix}

    Your solutions are:

    x_{1}=\frac{-2}{3}t\;\text{and}\;\  x_{2}=t

    x_{1}=t\;\text{and}\;\  x_{2}=t, respectively.

    Bases for the eigenspace are:

    \begin{bmatrix}\frac{-2}{3}\\1\end{bmatrix}

    \begin{bmatrix}1\\1\end{bmatrix}

    Let this be P:

    \Large{P=\begin{bmatrix}\frac{-2}{3}&1\\1&1\end{bmatrix}}

    \Large{P^{-1}=\begin{bmatrix}\frac{-3}{5}&\frac{3}{5}\\\frac{3}{5}&\frac{2}{5}\end{bma  trix}}

    Now, find P^{-1}AP. You have all the necessary info.

    Check me out. Easy to flub up.

    I hope this helps.

    If you don't have one, get a good linear algebra text. Anton and Rorres publish some good ones.
    Last edited by galactus; July 13th 2006 at 09:11 AM.
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  3. #3
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    I am completely confused on this whole problem. If you could finish this off, then put an example for me to do and ill do it and post it to see if I got it right, thanks.
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  4. #4
    Eater of Worlds
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    I went ahead and made some changes. I hope you can finish up. The worst is done.

    Good luck on your distance learning class. I know they can be a booger.

    Try this one:

    Find a matrix P that diagonalizes:

    A=\begin{bmatrix}0&0&-2\\1&2&1\\1&0&3\end{bmatrix}
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