The supports of $\varphi_1$ and $\varphi_2$ are disjoint, so they are <orthogonal vectors>. Substitution in the covariance formula gives
$$
C\varphi_1
=\frac12\|\varphi_1\|^2\varphi_1
=\frac{\lambda_1(A)}2\varphi_1,
$$
and
$$
C\varphi_2
=\frac12\|\varphi_2\|^2\varphi_2
=\frac{1-\lambda_1(A)}2\varphi_2.
$$
Therefore the associated <eigenvalue>[eigenvalues] are
$$
\boxed{\gamma_1=\frac{\lambda_1(A)}2,
\qquad
\gamma_2=\frac{1-\lambda_1(A)}2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 4 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-326/4/2/c/solution