Types of idempotent matrices $A^2 = A$
Can I classify some types of matrices as idempotent ?
The trivial examples are $0$ and $I_n$, and I've also found that a matrix with one row all equal to $1$ and every other row equal to $0$ is idempotent.
[Math] Types of idempotent matrices $A^2 = A$
linear algebra
Best Answer
Such matrices are projection operators on $\mathbb{F}^n$, and are always diagonalizable. The matrix $A$ is then similar to Dg$[I_r,0]$ where $r$ is the rank of $A$, that is, you can find an invertible matrix $P$ so that $P^{-1}AP=$ Dg$[I_r,0]$.
In this particular case we always have that $\mathbb{F}^n$ is the direct sum of the nullspace of $(A)$ and the column space of $A$, and we say that $A$ is the projection of $\mathbb{F}^n$ onto the column space of $A$ along the nullspace of $A$.
Furthermore if the column space of $A$ is the orthogonal complement of the nullspace of $A$ then $A$ is an orthogonal projection, and you can find a unitary matrix $P$ so that $P^{*}AP=$ Dg$[I_r,0]$. So we can say if $A$ is not an orthogonal projection, it is an oblique projection.