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Thread: Matrix help please

  1. #1
    Junior Member rednest's Avatar
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    Cool Matrix help please

    The transformation $\displaystyle R$ is represented by the matrix A, where A = $\displaystyle \left[ \begin{array}{ c c } 3 & 1 \\ 1 & 3 \end{array} \right]$

    (1) Find the eigenvectors of A.
    (2) Find an orthogonal matrix P and a diagonal matrix D such that $\displaystyle A = PDP^{-1}$
    (3) Hence describe the transformation $\displaystyle R$ as a combination of geometrical transformations, stating clearly their order.

    I got the answer for question (1) and (2), but I don't get question (3).
    Please help! My exam is next week. Thanks.
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  2. #2
    MHF Contributor Reckoner's Avatar
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    Quote Originally Posted by rednest View Post
    (3) Hence describe the transformation $\displaystyle R$ as a combination of geometrical transformations, stating clearly their order.
    For number 2, you should have got

    $\displaystyle A = PDP^{-1} = PDP^T$

    with

    $\displaystyle P = \left[\begin{matrix}
    \frac{\sqrt2}2 & \frac{\sqrt2}2\\
    -\frac{\sqrt2}2 & \frac{\sqrt2}2
    \end{matrix}\right]$

    $\displaystyle D = \left[\begin{matrix}
    2 & 0\\
    0 & 4
    \end{matrix}\right]$

    (it's okay if you did your eigenvalues and eigenvectors in a different order though, as long as $\displaystyle P$ is orthogonal)

    So we have

    $\displaystyle A = \left[\begin{matrix}
    \frac{\sqrt2}2 & \frac{\sqrt2}2\\
    -\frac{\sqrt2}2 & \frac{\sqrt2}2
    \end{matrix}\right]\left[\begin{matrix}
    2 & 0\\
    0 & 4
    \end{matrix}\right]\left[\begin{matrix}
    \frac{\sqrt2}2 & -\frac{\sqrt2}2\\
    \frac{\sqrt2}2 & \frac{\sqrt2}2
    \end{matrix}\right]$

    $\displaystyle = \left[\begin{matrix}
    \cos\left(-\frac\pi4\right) & -\sin\left(-\frac\pi4\right)\\
    \sin\left(-\frac\pi4\right) & \cos\left(-\frac\pi4\right)
    \end{matrix}\right]\left[\begin{matrix}
    2 & 0\\
    0 & 4
    \end{matrix}\right]\left[\begin{matrix}
    \cos\left(\frac\pi4\right) & -\sin\left(\frac\pi4\right)\\
    \sin\left(\frac\pi4\right) & \cos\left(\frac\pi4\right)
    \end{matrix}\right]$

    $\displaystyle = \left[\begin{matrix}
    \cos\left(-\frac\pi4\right) & -\sin\left(-\frac\pi4\right)\\
    \sin\left(-\frac\pi4\right) & \cos\left(-\frac\pi4\right)
    \end{matrix}\right]\left[\begin{matrix}
    2 & 0\\
    0 & 1
    \end{matrix}\right]\left[\begin{matrix}
    1 & 0\\
    0 & 4
    \end{matrix}\right]\left[\begin{matrix}
    \cos\left(\frac\pi4\right) & -\sin\left(\frac\pi4\right)\\
    \sin\left(\frac\pi4\right) & \cos\left(\frac\pi4\right)
    \end{matrix}\right]$

    Can you take it from here?
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