Correction to: Lie–Poisson Methods for Isospectral Flows (Foundations of Computational Mathematics, (2020), 20, 4, (889-921), 10.1007/s10208-019-09428-w)

Motivation The authors of [1] have found an error in the proof of Proposition 1. Specifically, the proof presented there implicitly assumed that the matrices ∇H(Q(t)†P(t))† commute for any t≥0. The statement of the Proposition 1 has now been reformulated in a suitable way to fit with the discussion below Proposition 1 in [1]. Here below, we present the corrected version and proof of Proposition 1. Consider a solution (Q(t), P(t)) of Hamilton’s equations (12) defined for 0≤t≤T and a given initial point (Q0,P0), such that Q0†P0∈S, for S⊆gl(n,C)∗ a linear subspace as before, and Q0 is invertible. Then, there exists G∈GL(n,C) such that Q(t)†G∈N(S), for any 0≤t≤T,1 if and only if ∇H(Q(t)†P(t))†∈n(S), for any 0≤t≤T. Furthermore, if one of the two condition holds, then Q†(t)P(t)∈S, for any 0≤t≤T. Let us first prove the equivalence of the two conditions. Let us assume that ∇H(Q(t)†P(t))†∈n(S), for any 0≤t≤T. We have (Formula presented.) where Θ is a solution to (Formula presented.) where dexp-1 is defined as in [2]. Hence, Θ(t)∈n(S), for any 0≤t≤T, being defined as the Magnus expansion of ∇H(Q(t)†P(t))† (see [2]). This proves the statement, since N(S)⊇exp(n(S)). Conversely, let us assume that there exists G∈GL(n,C) such that Q(t)†G∈N(S) for any 0≤t≤T. By the formula above, we have (Formula presented.) Since the left-hand side is in N(S) for any 0≤t≤T, we have Θ(t)∈n(S). This implies that also dΘ(t)dt∈n(S). Therefore, since dexpΘ(t)dΘ(t)dt=∇H(Q†P)†, we get the thesis. Finally, we have (Formula presented.) Hence, if one of the two condition holds, Q†(t)P(t)∈S, for any 0≤t≤T, by the definition of N(S). □