= Solution
For an <implicit Runge-Kutta method>, <algebraic stability> means that $b_i\geq0$ and the symmetric matrix
$$
\mathcal M_{ij}=b_i a_{ij}+b_j a_{ji}-b_i b_j
$$
is <positive semidefinite>. Here $b_1=b_2=1/2$ and $A+A^T=ee^T/2$, so
$$
\mathcal M=\frac12(A+A^T)-\frac14ee^T=0.
$$
\b[The method is algebraically stable.] This also explains why the property matters for nonlinear problems. If a vector field satisfies $\langle f(v)-f(w),v-w\rangle\leq0$, the <Runge-Kutta contractivity identity> for two solutions, with stage differences $D_i$ and vector-field differences $F_i$, is
$$
\|d_{n+1}\|^2-\|d_n\|^2
=2h\sum_i b_i\langle D_i,F_i\rangle
-h^2\sum_{i,j}\mathcal M_{ij}\langle F_i,F_j\rangle\leq0.
$$
The first sum is nonpositive and the second vanishes. Thus, whenever the implicit stage equations have the relevant solutions, their updates are contractive in the <Hilbert space> norm.
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