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?
Best Answer
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