information:highlighting, information:classification, stylistic:style_transfer
Orthogonal projection

For example, the function which maps the point (x, y, z) {\displaystyle (x,y,z)} in three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} to the point (x, y, 0) {\displaystyle (x,y,0)} is an orthogonal projection onto the xy -plane. This function is represented by the matrix P =. {\displaystyle P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}.}

The action of this matrix on an arbitrary vector is P =. {\displaystyle P{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}.}

To see that P {\displaystyle P} is indeed a projection, i.e., P = P 2 {\displaystyle P=P^{2}}, we compute P 2 = P = = P. {\displaystyle P^{2}{\begin{bmatrix}x\\y\\z\end{bmatrix}}=P{\begin{bmatrix}x\\y\\0\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}=P{\begin{bmatrix}x\\y\\z\end{bmatrix}}.}

Observing that P T = P {\displaystyle P^{\mathrm {T} }=P} shows that the projection is an orthogonal projection.