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Math Help - Matrix help please

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

    The transformation R is represented by the matrix A, where A = \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 A = PDP^{-1}
    (3) Hence describe the transformation 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 R as a combination of geometrical transformations, stating clearly their order.
    For number 2, you should have got

    A = PDP^{-1} = PDP^T

    with

    P = \left[\begin{matrix}<br />
\frac{\sqrt2}2 & \frac{\sqrt2}2\\<br />
-\frac{\sqrt2}2 & \frac{\sqrt2}2<br />
\end{matrix}\right]

    D = \left[\begin{matrix}<br />
2 & 0\\<br />
0 & 4<br />
\end{matrix}\right]

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

    So we have

    A = \left[\begin{matrix}<br />
\frac{\sqrt2}2 & \frac{\sqrt2}2\\<br />
-\frac{\sqrt2}2 & \frac{\sqrt2}2<br />
\end{matrix}\right]\left[\begin{matrix}<br />
2 & 0\\<br />
0 & 4<br />
\end{matrix}\right]\left[\begin{matrix}<br />
\frac{\sqrt2}2 & -\frac{\sqrt2}2\\<br />
\frac{\sqrt2}2 & \frac{\sqrt2}2<br />
\end{matrix}\right]

    = \left[\begin{matrix}<br />
\cos\left(-\frac\pi4\right) & -\sin\left(-\frac\pi4\right)\\<br />
\sin\left(-\frac\pi4\right) & \cos\left(-\frac\pi4\right)<br />
\end{matrix}\right]\left[\begin{matrix}<br />
2 & 0\\<br />
0 & 4<br />
\end{matrix}\right]\left[\begin{matrix}<br />
\cos\left(\frac\pi4\right) & -\sin\left(\frac\pi4\right)\\<br />
\sin\left(\frac\pi4\right) & \cos\left(\frac\pi4\right)<br />
\end{matrix}\right]

    = \left[\begin{matrix}<br />
\cos\left(-\frac\pi4\right) & -\sin\left(-\frac\pi4\right)\\<br />
\sin\left(-\frac\pi4\right) & \cos\left(-\frac\pi4\right)<br />
\end{matrix}\right]\left[\begin{matrix}<br />
2 & 0\\<br />
0 & 1<br />
\end{matrix}\right]\left[\begin{matrix}<br />
1 & 0\\<br />
 0 & 4<br />
 \end{matrix}\right]\left[\begin{matrix}<br />
\cos\left(\frac\pi4\right) & -\sin\left(\frac\pi4\right)\\<br />
\sin\left(\frac\pi4\right) & \cos\left(\frac\pi4\right)<br />
\end{matrix}\right]

    Can you take it from here?
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