linear-guest

14.4 Gram-Schmidt & Orthogonal Complements 261
We can check that P T P = I by a lengthy computation, or more simply, notice
that
(P T P ) =
=
⎛
⎝
u T 1
u T 2
u T 3
⎞
⎠ ( u 1 u 2 u 3
)
⎛
1 0
⎞
0
⎝0 1 0⎠ .
0 0 1
Above we are using orthonormality of the u i and the fact that matrix multiplication
amounts to taking dot products between rows and columns. It is also very important
to realize that the columns of an orthogonal matrix are made from an orthonormal
set of vectors.
Orthonormal Change of Basis and Diagonal Matrices. Suppose D is a diagonal
matrix and we are able to use an orthogonal matrix P to change to a new basis. Then
the matrix M of D in the new basis is:
Now we calculate the transpose of M.
The matrix M = P DP T is symmetric!
M = P DP −1 = P DP T .
M T = (P DP T ) T
= (P T ) T D T P T
= P DP T
= M
14.4 Gram-Schmidt & Orthogonal Complements
Given a vector v and some other vector u not in span {v} we can construct
the new vector
v ⊥ := v − u · v u · u u .
261