= 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 $F_i=f(t_n+c_i h,Y_i)-f(t_n+c_i h,Z_i)$. Then
$$
D_i=d+h\sum_j a_{ij}F_j,\qquad
d_+=d+h\sum_i b_iF_i.
$$
Expanding the squared <norm> of $d_+$ gives
$$
\|d_+\|^2=\|d\|^2+2h\sum_i b_i\langle d,F_i\rangle
+h^2\sum_{i,j}b_i b_j\langle F_i,F_j\rangle.
$$
In each term of the linear sum, replace $d$ by $D_i-h\sum_ja_{ij}F_j$. Symmetrizing the resulting double sum proves the <Runge-Kutta contractivity identity>:
$$
\boxed{\|d_+\|^2-\|d\|^2
=2h\sum_i b_i\langle D_i,F_i\rangle
-h^2\sum_{i,j}m_{ij}\langle F_i,F_j\rangle.}
$$
Dissipativity makes $\langle D_i,F_i\rangle\leq0$, and the weights are nonnegative. For the second term, positive semidefiniteness permits a factorization $M=C^TC$, giving
$$
\sum_{i,j}m_{ij}\langle F_i,F_j\rangle
=\sum_k\left\|\sum_i C_{ki}F_i\right\|^2\geq0.
$$
For $h\geq0$ both contributions in the distance-change identity are nonpositive. Therefore \b[$\|y_{n+1}-z_{n+1}\|\leq\|y_n-z_n\|$]. 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.
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