# Why is the product of these projection matrices a zero matrix

2 vectors are given: $$\vec{a_1}=\begin{bmatrix}1 && 2 && -2\end{bmatrix}^T \\ \vec{a_2}=\begin{bmatrix}-2 && 2 && 1\end{bmatrix}^T$$

To calculate their projection matrices, I used the formula: $$P = \frac{\vec{a}{\vec{a^T}}}{\vec{a^T}{\vec{a}}}$$

and got the following matrices:

$$\vec{P_1}=\begin{bmatrix}1 && 2 && -2 \\ 2 && 4 && -4 \\ -2 && -4 && 4 \end{bmatrix} \\ \vec{P_2}=\begin{bmatrix}4 && -4 && -2 \\ -4 && 4 && 2 \\ -2 && 2 && 1\end{bmatrix}$$

And their product is a zero matrix, but I can't explain why.

Is it because the vectors are orthogonal?

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yes. $P_1P_2= a_1a_1^Ta_2a_2^T= a_10a_2^T=0$.Here by $0$ I mean zero matrix of the corresponding dimension. – Alexander Vigodner Jun 20 '14 at 15:33

Notice that the two vectors are orthogonal so the kernel of the projection $P_1$ is the image of the projection $P_2$ and then their product is the zero matrix
$P_1$ is the orthogonal projection onto $\operatorname{span}(a_1)$ so it's the image of $P_1$ and $\operatorname{span}(a_2)$ is the kernel since $a_1\perp a_2$. – user63181 Jun 20 '14 at 15:56