The authors consider the problem of maximizing the functional \[ f(U)= \sum^n_{i=1} \max_j \{(U^T M_jU)_{ii}\} \] over all \(n\times n\) orthogonal matrices \(U\) and show that any orthogonal matrix \(Q\) which simultaneously diagonalizes the \(M_j\) maximizes \(f\).