Matrix help please

**rednest** · June 12th 2008, 10:53 PM

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.

**Reckoner** · June 13th 2008, 12:00 AM

Originally Posted by rednest

(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

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

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

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

So we have

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

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

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

Can you take it from here?

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