Chapter 7.5, Problem 6E

[M] The covariance matrix below was obtained from a Land-sat image of the Columbia River in Washington, using data from three spectral bands. Let x₁, x₂, x₃ denote the spectral components of each pixel in the image. Find a new variable of the form y₁= c₁x₁ + c₂x₂ + c₃x₃ that has maximum possible variance, subject to the constraint that $c_1^2+c_2^2+c_3^2=1$.What percentage of the total variance in the data is explained by y₁?

$S=\left[\begin{array}{ccc}29.64& 18.38& 5.00\\ 18.38& 20.82& 14.06\\ 5.00& 14.06& 29.21\end{array}\right]$