Exponential of an idempotent linear map
= Exponential of an idempotent linear map
{title2=$e^{\alpha P}=(I-P)+e^\alpha P$}
For an <idempotent linear map> $P$, induction gives $P^j=P$ for all integers $j\geq1$. Summing the <matrix exponential> series therefore proves the displayed identity. The <direct sum> of its <kernel of a linear map> and <image of a linear map> is fixed and scaled by $e^\alpha$, respectively. The result requires no <Hermitian matrix> hypothesis and is valid for real or complex $\alpha$.