What does it mean by a matrix is a projector on some subspace?

Question

Could anyone explain to me what does mathematically mean a matrix $A$ is a projector on some subspace $V\subset\mathbb R^n$ in general?

Does it mean $(A^2-A)x=0\forall x\in V$?

Thankx.

Answer 1

First of all, "$A$ is a projection" means that $A^2 = A$. (This means that $(A^2 - A)x = 0$ for all $ x \in \mathbb R^n$, not just for $x$ taken from some subspace.)

And the statement that "$A$ is a projection onto the subspace $V \subset \mathbb R^n$" is equivalent to saying that $A$ is a projection and the image of $A$ is $V$.

In general, if $A$ is a projection, and if $U : = {\rm ker}(A)$ and $V : = {\rm im}(A)$, then $\mathbb R^n$ is the direct sum $ U \oplus V$. In other words, for every element $x \in \mathbb R^n$ there exist $u \in U, v \in V$ such that $x = u + v$, and the choice of $u$ and $v$ is unique for this $x$. In terms of this decomposition, the action of $A$ on $x$ is given by $A(x) = v$. It is in this sense that linear transformation $A$ "projects" the vector $x$ "onto" the subspace $V$.