Are idempotent matrices always a projection matrix?

Question

I know that a projection matrix is always an idempotent matrix, but is it true that a idempotent matrix is always a projection matrix?

Answer 1

If $M \in Mat_{n \times n}(F)$ is an idempotent matrix, then you can show that $F^n = \operatorname{Im}(M) \oplus \ker(M)$, and $M$ is the projection onto its image along the kernel.

Note that this is not an orthogonal projection in general.

Answer 2

As far as I know, the terms are synonyms (when applied to matrices.)