On eigenvalues of a matrix arising in energy-preserving/dissipative continuous-stage Runge-Kutta methods

In this short note, we define an s × s matrix Ks constructed from the Hilbert matrix Hs=(1i+j-1)i,j=1s{H_s} = \left( {{1 \over {i + j - 1}}} \right)_{i,j = 1}^s and prove that it has at least one pair of complex eigenvalues when s ≥ 2. Ks is a matrix related to the AVF collocation method, which is an energy-preserving/dissipative numerical method for ordinary differential equations, and our result gives a matrix-theoretical proof that the method does not have large-grain parallelism when its order is larger than or equal to 4.