Linear Algebra and Its Applications · Problem 3.3.16

Orthogonality — Gilbert Strang

Mathematics · rank-one-projection-matrix

Problem

Show that if

u

has unit length, then the rank-1 matrix

P = uu^T

is a projection matrix: It has properties (i) and (ii) in 3N. By choosing

u = a/\|a\|

P

becomes the projection onto the line through

a

, and

Pb

is the point

p = \hat{x}a

. Rank-1 projections correspond exactly to least-squares problems in one unknown.