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Any rank-1 matrix $A$ can be written as $A = c r^T$ where $c$, $r$ are both in $\mathbb R^n$. $A$ can also be considered a projection matrix projecting any vector $w$ onto $c$. $AA =(c r^T)(c r^T) = r^Tc (c r^T)=r^Tc A$. However if $c$ and $r$ are orthogonal then $A$ projects any vector $w$ onto $c$ except a vector already in the subspace of $c$ in which case $A c = 0$. Is $A$ still a projection matrix?

Hagen von Eitzen
  • 374,180
Tom Agnew
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1 Answers1

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In general no (and if it is, it is not of rank 1). If $A$ is a projection matrix then $AA=A$. And if $r^Tc=0$ then $AA=0$. So the only way for $A$ to still be a projection matrix is when $A=0$, but then its rank is not $0$. Actually, $cr^T$ is a rank 1 projection matrix only when $r^Tc=1$.

Hagen von Eitzen
  • 374,180