Past exam of the algorithmic foundations of learning course of the University of Oxford / 2025 / 2 / a / ii / Solution

Let $a_1,\dots ,a_d>0$ be the eigenvalues of the positive-definite matrix $V_n$. Its trace satisfiesThe arithmetic-geometric mean inequality givesSince $\det V_0={\lambda }^d$, taking the logarithm provesThe determinant is largest, for a fixed trace, when the eigenvalues are equal; this is the geometric content of the estimate.