= Solution
Define the stage states $Y_i=y_n+h\sum_j a_{ij}k_j$, so $k_i=PY_i$. Skew symmetry gives $Y_i^Tk_i=Y_i^TPY_i=0$, and hence
$$
y_n^Tk_i=-h\sum_j a_{ij}k_j^Tk_i.
$$
Insert this into the preceding norm identity and symmetrize the double sum, using $k_i^Tk_j=k_j^Tk_i$. The result is the <Runge-Kutta conservation of quadratic invariants> identity
$$
\lVert y_{n+1}\rVert^2-\lVert y_n\rVert^2=-h^2\sum_{i,j}\left(b_i a_{ij}+b_j a_{ji}-b_i b_j\right)k_i^Tk_j.
$$
Thus
$$
\boxed{M=0\quad\Longrightarrow\quad\lVert y_{n+1}\rVert=\lVert y_n\rVert,}
$$
which proves the sufficient condition for every well-defined step and then for all $n$ by induction. For example the <implicit midpoint rule>, with $a_{11}=1/2$ and $b_1=1$, has $M_{11}=0$ and preserves this norm exactly. The condition is sufficient, and need not be necessary for a particular matrix or a particular solution.
Back to article page
