Question

Let A be the 3 x 3 matrix such that, for any. ✓ E R', Av gives the projection of ū onto the plane x + y + z = 0. Determine A15

Answer 1

Since $A$ is a projection matrix, so, $A^2=A$ and $A=A^T$

$\Rightarrow A^{15}=A$ by induction.

So, it is enough to find the matrix $A$.

Now,

$\begin{pmatrix} \frac{1}{\sqrt2}\\ 0 \\\frac{-1}{\sqrt2} \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{\sqrt6}\\ \frac{-2}{\sqrt6} \\\frac{1}{\sqrt6} \end{pmatrix}$ form an orthonormal basis for the plane $x+y+z=0$

Therefore, the matrix $A$ is given by

$A=\begin{pmatrix} \frac{1}{\sqrt2}&\frac{1}{\sqrt6}\\ 0&\frac{-2}{\sqrt6} \\\frac{-1}{\sqrt2}&\frac{1}{\sqrt6} \end{pmatrix}\begin{pmatrix} \frac{1}{\sqrt2}&0&\frac{-1}{\sqrt2}\\ \frac{1}{\sqrt6}&\frac{-2}{\sqrt6}&\frac{1}{\sqrt6} \end{pmatrix}=\begin{pmatrix} \frac{2}{3} & \frac{-1}{3}&\frac{-1}{3} \\ \frac{-1}{3} &\frac{2}{3} &\frac{-1}{3} \\ \frac{-1}{3}& \frac{-1}{3} & \frac{2}{3} \end{pmatrix}$