Consider a random vector
Z with mean and covariance matrix
[3.69]
a. Calculate the determinant of the corresponding
correlation matrix.
b. Is Z singular, multicollinear, or neither
of these?
c.
Calculate the eigenvalues and eigenvectors of
.
d. Represent Z in terms of its principal
components as in [3.66].
e. What is the covariance matrix
of the vector of principal components D?
f. Construct an approximation
for Z based on the first two principal
components of Z.
g. Construct the covariance matrix of
.
Compare your result with the covariance matrix of Z.
Solution
a. The correlation matrix has determinant
= .009314.
b. Based upon item (a), Z is multicollinear.
c. The eigenvalues
are 33.0548, 15.6856 and 0.0759 respectively. The corresponding eigenvectors
form the columns of
[s1]
d. The principal components of Z are the
normalized eigenvectors of the covariance matrix
.
The eigenvectors we presented for item (c) are normalized, so they are the
principal components of Z.
_small.jpg)
Calculate the eigenvalues and eigenvectors of
.
of the vector of principal components D? 
for Z based on the first two principal
components of Z. 
= .009314. 
are 33.0548, 15.6856 and 0.0759 respectively. The corresponding eigenvectors
form the columns of
