Given a vector $v\in \mathbb{R}^n$, I have seen from several sources that $I-vv^T$ is the projection onto the orthogonal complement of the line through the origin in the direction $v$?
I have unfortunately memorized this expression instead of figuring out what it actually means. I know that the orthogonal complement of a line is a plane with the line being its normal vector. And from what I understand this means that $P=I-vv^T$ is a projection matrix which places vectors into that aforementioned plane. But beyond that, I'm having difficulty connecting $I-vv^T$ to all vectors which are orthogonal to $v$ (which is what an orthogonal complement is, after all).
Can someone help me understand how this is true?
Best Answer
Given a vector $x$, and a unit vector $v$, the projection of $x$ onto $v$ is going to be $(v^Tx)v=v(v^Tx)=(vv^T)x$.
Hence the orthogonal complement is $x-(v^Tx)v=x-(vv^T)x=(I-vv^T)x.$