Every matrix \(A\) such that \(A^2 = A\) is similar to a diagonal matrix.
Solution
Let \(T\) be the operator represented by \(A\), so \(T^2=T\). For every \(v\), \[ v=(v-Tv)+Tv, \] with \[ T(v-Tv)=Tv-T^2v=0, \] so \(v-Tv\in\ker T\), while \(Tv\in\operatorname{im}T\). Thus \[ V=\ker T+\operatorname{im}T. \] If \(w\in\ker T\cap\operatorname{im}T\), write \(w=Tv\). Then \[ w=Tv=T^2v=Tw=0, \] so the sum is direct. Choose bases of \(\ker T\) and \(\operatorname{im}T\); their union is a basis of \(V\). In that basis, \(T\) is diagonal with only \(0\)’s and \(1\)’s on the diagonal. Hence \(A\) is similar to a diagonal matrix.