Projections
Contents
Definition
Given two vectors a and b, we can project b onto a to get the best possible estimate of the former as a multiple of the latter. This projection p has an error term e.
Take the multiple as x, so that p = ax. The error term can be characterized as b-ax.
a is orthogonal to e. Therefore, aT(b-ax) = 0. This simplifies to x = (aTb)/(aTa). Altogether, the projection is characterized as p = a(aTb)/(aTa).
A matrix P can be defined such that p = Pb. The projection matrix is (aaT)/(aTa).