Page 434 - Matrix Analysis & Applied Linear Algebra

430 Chapter 5 Norms, Inner Products, and Orthogonality

T
T
⊥
observe that if N = M , then N M = 0 and M N = 0. Furthermore, if

T
T
dim M = r, then M M is r × r, and rank M M = rank (M)= r by
T
(4.5.4), so M M is nonsingular. Therefore, if the columns of N are chosen to
⊥
be an orthonormal basis for M , then
 −1   −1 
T
T
M M M T M M M T
I 0 −1
  M | N = 0 I =⇒ M | N =   .
N T N T
This together with (5.13.1) says the orthogonal projector onto M is given by
 −1 
T
M M M T
T −1 T
P M = M | 0   = M M M M . (5.13.2)
N T
As discussed in §5.9, the projector associated with any given pair of com-
plementary subspaces is unique, and it doesn’t matter which bases are used to
−1
T
form M and N in (5.13.1). Consequently, formula P M = M M M M T
is independent of the choice of M —just as long as its columns constitute some
basis for M. In particular, the columns of M need not be an orthonormal basis
T
T
for M. But if they are, then M M = I, and (5.13.2) becomes P M = MM .
Moreover, if the columns of M and N constitute orthonormal bases for M and

M , respectively, then U = M | N is an orthogonal matrix, and (5.13.1) be-
⊥
comes

I r 0 T
P M = U U .
0 0
In other words, every orthogonal projector is orthogonally similar to a diagonal
matrix in which the diagonal entries are 1’s and 0’s.
Below is a summary of the formulas used to build orthogonal projectors.
Constructing Orthogonal Projectors
n
Let M be an r-dimensional subspace of , and let the columns of
M n×r and N n×n−r be bases for M and M , respectively. The or-
⊥
⊥
thogonal projectors onto M and M are
T −1 T T −1 T
• P M = M M M M and P M ⊥ = N N N N . (5.13.3)
If M and N contain orthonormal bases for M and M , then
⊥
T
• P M = MM T and P M ⊥ = NN . (5.13.4)
0
T
• P M = U I r U , where U = M | N . (5.13.5)
0 0
• P M ⊥ = I − P M in all cases. (5.13.6)
Note: Extensions of (5.13.3) appear on p. 634.