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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.

The Butcher contractivity theorem states that an algebraically stable Runge-Kutta method is B-stable: for a dissipative vector field and any positive step for which the stage problems are solved, its step does not increase the distance between two numerical solutions. The required condition on the dissipative vector field isfor all compared states at each common time. The inner product can be real Euclidean or complex Hermitian. The stage solvability of an implicit Runge-Kutta method is assumed in this comparison, not a conclusion from a distance estimate alone.

Let $d$ be the difference of starting values, $D_i$ the differences of corresponding stages and $F_i$ the differences of their vector-field values. ThenExpanding the squared norm of the update givesSubstitute $d=D_i-h{\sum }_j{a}_{ij}{F}_j$ in the linear term. Symmetrizing the double sum proves the Runge-Kutta contractivity identityEach inner product in the first sum is nonpositive by the defining inequality for a dissipative vector field, and its weight is nonnegative. The final quadratic form is nonnegative: resolve the vectors $F_i$ into components and apply positive semidefiniteness of $M$ to each component vector. Hence $\boxed{\Vert d_{+}\Vert \le \Vert d\Vert }$, which is exactly B-stability. No linearization of the vector field has been used.

For the Dahlquist test equation $y^{\prime }=\lambda y$ with $\mathrm{Re}\lambda \le 0$, this also bounds the amplification factor by one wherever the stages are well defined. With strictly positive weights, no stage pole can occur in the open left half-plane. Indeed, if $(I-zA)v=0$ for nonzero $v$, thenwhen $\mathrm{Re}z<0$, contradicting $M⪰0$. This proves the usual A-stability corollary for the scalar stability function; boundedness from the open half-plane makes any boundary pole in that rational function removable. Zero-weight redundant stages require their own solvability treatment. The nonlinear B-stability statement is stronger than A-stability, and the two notions should not be conflated.