STAT312 Lecture : Orthogonality; projections.pdf

• School: University of Alberta
• Department: Statistics
• Course: STAT312
• Professor: Douglas Wiens

Document Summary

Here we introduce a matrix the hat matrix of special importance in regression. It will also be used to motivate some of what follows. Consider a regression model y = x + with x of full rank . We will later show that the lses are. [y] = x is y = x = X0 is the hat matrix it places the hat on y . Hx = x (i h)x = 0 (i h)2 = (i h) Ned by between non-zero vectors x y is de- cos = x0y kxk kyk equivalent to the statement that x0y (what is this in r2?) That such an angle exists is kxk kyk. This in turn is a version of the famous cauchy- Proof of this version: for any real number. 0 kx + yk2 = kyk2 2 + 2x0y + kxk2 so that there is at most one real zero.