= Powers of a rank-one operator
{title2=$K^m=\varphi(a)^{m-1}K\quad(m\geq1)$}
Let $Kx=a\varphi(x)$ on a <vector space>, where $\varphi$ is a <linear functional>. Its <image of a linear map> lies in the span of $a$. Applying $K$ twice gives $K^2x=a\varphi(a)\varphi(x)=\varphi(a)Kx$. <Mathematical induction> then proves the displayed power formula. If $\varphi(a)=0$, $K^2=0$, even when $K\ne0$; this is a <nilpotent operator>. If $\varphi(a)\ne0$, then $K/\varphi(a)$ is an <idempotent linear map>. In a real <inner product space>, the special case $\varphi(x)=b\cdot x$ gives $\varphi(a)=a\cdot b$.
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