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).

2. Originally Posted by rednest
(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?