Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 3 / d / Solution

The eigenvector $(1,-1,0{)}^{\mathsf{T}}$ has eigenvalue $2$. On its orthogonal complement, write vectors as $(a,a,c{)}^{\mathsf{T}}$. The eigenvalue equation becomesHence the ordered eigenvalues of the covariance matrix are $4+2\sqrt{2}$, $2$, and $4-2\sqrt{2}$. The first two principal components explainof the total variance. The denominator is the sum of all eigenvalues, which equals the trace.