Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 205 / 6 / a

Taking $\Theta =0$ gives $\Vert \Sigma \Theta -I{\Vert }_{\max }=1$, so every matrix is $1$-invertible. The smallest admissible value isThe map inside the norm is affine in $\Theta$, and a norm composed with an affine map is a convex function. The space of matrices is a convex feasible set, so this is a convex optimization problem.