Question about the orthogonal projection

Question

Is projection matrix and orthogonal projection matrix means same or they are different?

Answer 1

In general, a projection matrix is any matrix $P$ that satisfies $P^2=P$. All orthogonal projections satisfy this property, but there are indeed projections which are not orthogonal.

The easiest way to see this is perhaps geometryically. Consider, say, the projection on $\mathbb{R}^2$ onto the $x$-axis that is parallel to the line $y=x$.

Answer 2

A projection is orthogonal if and only if it is self-adjoint. If a projection matrix is not orthogonal, it is sometimes called an oblique projection.