A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

52 PCA and SVD

directions of extremal variance are pairwise orthogonal and form

an orthonormal basis for the space R

d

. Let us name the eigenval-

ues of S, λ

1

, λ

2

, . . . , λ

d

, and the eigenvectors v

1

, v

2

, . . . , v

d

. The

extremal values of the variance are, therefore,

hSv

i

, v

i

i = hλ

i

v

i

, v

i

i = λ

i

hv

i

, v

i

i = λ

i

.

This means that we can sort the extremal directions v

i

in the order

of importance simply by sorting them according to the correspond-

ing eigenvalues. In particular, the direction of maximal variance

corresponds to the eigenvector with the largest eigenvalue, and the

direction of minimal variance corresponds to the eigenvector with

the smallest eigen ...