We know a square matrix $A$ that gives vector space projection from $R^n$ to its subspace, say, $V$ is indeed a projection matrix if $A^2 = A$.
If I had, for example, a square matrix $B$ which satisfies $B^3 = B$:
- Would $B$ be still considered a valid projection from $R^n$ to a subspace, say, $W$?
- If not, would $B^2$ be a projection from $R^n$ to its own corresponding subspace?
Thank you in advance for any commentary.