Let $A$ be an $n\times n$ matrix such that $A^2=A$.
a) Let $E_{1}(A)=\{x \in \mathbb{R^n} | Ax=x \}$: let $E_{0}(A)=\{{ x \in \mathbb{R^n} | Ax=0\}}$. Let $x$ be any vector in $\mathbb{R^n}$. Show that $Ax \in E_{1}(A)$ and $x-Ax \in E_{0}(A)$
b) Show that if A is diagonalizable, then rank$(A)=tr(A)$
There are the last two problems in my problem set that have me absolutely stumped. Could someone be kind enough to provide solutions and a brief explanation? Thanks.
Part a) is simply a matter of applying definitions. What can we say about $A(Ax)$? What about $A(x - Ax)$? Remember that $A^2 = A$.
For part b), what could the eigenvalues of $A$ be? If $A$ is diagonalizable, it is similar to a diagonal matrix. What does that matrix look like? How do we compute its rank? Remember that two similar matrices have the same rank and trace.