 Solution

The stated hypotheses are precisely the algebraic criterion in part (a). To prove its stability content, compare two Runge-Kutta method steps for the same dissipative vector field. Let $d=y_n-z_n$ be the input difference, $D_i=Y_i-Z_i$ the stage differences, and $F_i=f(t_n+c_{i}h,Y_i)-f(t_n+c_{i}h,Z_i)$. ThenExpanding the squared norm of $d_{+}$ givesIn each term of the linear sum, replace $d$ by $D_i-h{\sum }_j{a}_{ij}{F}_j$. Symmetrizing the resulting double sum proves the Runge-Kutta contractivity identity:Dissipativity makes $⟨D_i,F_i⟩\le 0$, and the weights are nonnegative. For the second term, positive semidefiniteness permits a factorization $M=C^{T}C$, givingFor $h\ge 0$ both contributions in the distance-change identity are nonpositive. Therefore $\Vert y_{n+1}-z_{n+1}\Vert \le \Vert y_n-z_n\Vert$. This proves the contractivity assertion behind algebraic stability, usually called the Butcher contractivity theorem. It assumes the compared implicit stage equations have solutions; the matrix condition by itself should not be substituted for a separate stage-solvability argument.