Question

Let Z1=0.8X1+0.6X2 and Z2=−0.6X1+0.8X2 be the first and second principal components with corresponding eigenvalues λ1=3.82 and λ2=0.18. Then, var(Z1−3Z2)=5.44

true or false??

Answer 1

We know for principal components corresponding eigen value= variance explained.

Z1=0.8X1+0.6X2 , corresponding eigenvalue = 3.82. So Var ( Z1) = 3.82

and Z2=−0.6X1+0.8X2 , corresponding eigenvalue = 0.18. So Var ( Z2) = 0.18

var(Z1−3Z2) = var(Z1) + 9 * var(Z2) - 6 cov ( Z1,Z2) [ Z1 & Z2 are orthogonal so covariance 0 ]

= 3.82 + 9 * 0.18 = 5.44

So, the statement is true.

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