Chapter 7.5, Problem 9E

Suppose three tests are administered to a random sample of college students. Let X₁,…, X_N be observation vectors in ℝ³ that list the three scores of each student, and for j = 1,2,3, let x_j denote a student’s score on the jth exam.

Suppose the covariance matrix of the data is

$S=\left[\begin{array}{ccc}5& 2& 0\\ 2& 6& 2\\ 0& 2& 7\end{array}\right]$

Let y be an “index” of student performance, with y = c₁x₁ + c₂x₂ + c₃x₃ and $c_1^2+c_2^2+c_3^2=1$. Choose c₁,c₂,c₃ so that the variance of y over the data set is as large as possible. [Hint: The eigenvalues of the sample covariance matrix are λ = 3,6, and 9.]