Structure-Preserving Flows of Symplectic Matrix Pairs
This work provides a theoretical framework for structure-preserving flows in symplectic matrix pairs, which is relevant for numerical linear algebra and dynamical systems, but the results are theoretical and no concrete numerical improvements are demonstrated.
The authors construct a nonlinear differential equation for symplectic matrix pairs that preserves both the symplectic structure and invariant subspaces, governed by a Riccati differential equation. They use Grassmann manifolds to extend the flow's domain to the whole real line except isolated points.
We construct a nonlinear differential equation of matrix pairs $(\mathcal{M}(t),\mathcal{L}(t))$ that is invariant (the \textbf{Structure-Preserving Property}) in the class of symplectic matrix pairs \begin{align*} \mathbb{S}_{\mathcal{S}_1,\mathcal{S}_2}=\left\{\left(\mathcal{M},\mathcal{L}\right)| \ \mathcal{M}=\left[% \begin{array}{cc} X_{12} & 0 X_{22} & I \end{array}% \right]\mathcal{S}_2, \mathcal{L}=\left[% \begin{array}{cc} I & X_{11} 0 & X_{21} \end{array}% \right]\mathcal{S}_1\right.\nonumber \left. \text{ and }X=\left[% \begin{array}{cc} X_{11} & X_{12} X_{21} & X_{22} \end{array}% \right]\text{is Hermitian}\right\} \end{align*} for certain fixed symplectic matrices $\mathcal{S}_1$ and $\mathcal{S}_2$. Its solution also preserves invariant subspaces on the whole orbit (the \textbf{Eigenvector-Preserving Property}). Such a flow is called a \textit{structure-preserving flow} and is governed by a Riccati differential equation (RDE). In addition, Radon's lemma leads to an explicit form. Therefore, blow-ups for the structure-preserving flows may happen at a finite $t$. To continue, we then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole $\mathbb{R}$ subtracting some isolated points.