Notes on Principal Component Analysis

variables or factors, and the estimation of the factor structure). We
will get sloppy ourselves later, but some people really get exercised
about these things.
We will begin working with the population (but everything translates
more-or-less directly for a sample):
Suppose [X 1 , X 2 , . . . , X p ] = X ′ is a set of p random variables, with
mean vector µ and variance-covariance matrix Σ. I want to define p
linear combinations of X ′ that represent the information in X ′ more
parsimoniously. Specifically, find a 1 , . . . , a p such that a ′ 1X, . . . , a ′ pX
gives the same information as X ′ , but the new random variables,
a ′ 1X, . . . , a ′ pX, are “nicer”.
Let λ 1 ≥ λ 2 ≥ · · · ≥ λ p ≥ 0 be the p roots (eigenvalues) of the
matrix Σ, and let a 1 , . . . , a p be the corresponding eigenvectors. If
some roots are not distinct, I can still pick corresponding eigenvectors
to be orthogonal. Choose an eigenvector a i so a ′ ia i = 1, i.e., a
normalized eigenvector. Then, a ′ iX is the i th principal component of
the random variables in X ′ .
Properties:
1) Var(a ′ iX) = a ′ iΣa i = λ i
We know Σa i = λ i a i , because a i is the eigenvector for λ i ; thus,
a ′ iΣa i = a ′ iλ i a i = λ i . In words, the variance of the i th principal
component is λ i , the root.
Also, for all vectors b i such that b i is orthogonal to a 1 , . . . , a i−1 ,
and b ′ ib i = 1, Var(b ′ iX) is the greatest it can be (i.e., λ i ) when
b i = a i .
2