= Solution
For a <Runge-Kutta method> with real stage matrix $A=(a_{ij})$ and weights $b_i$, define
$$
M=\operatorname{diag}(b)A+A^T\operatorname{diag}(b)-bb^T,
\qquad m_{ij}=b_i a_{ij}+b_j a_{ji}-b_i b_j.
$$
The method has <algebraic stability of a Runge-Kutta method> if
$$
\boxed{b_i\geq0\text{ for every }i,\qquad M\text{ is positive semidefinite}.}
$$
The matrix condition means $z^TMz\geq0$ for every real vector $z$. Its significance is nonlinear contractivity: an algebraically stable method, on well-defined stage solutions, is <B-stable> for a <dissipative vector field> satisfying $\langle f(t,y)-f(t,z),y-z\rangle\leq0$. This is distinct from scalar <A-stability>, which concerns the linear test equation. The contractivity proof in part (b) explains the matrix criterion directly.
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