Annals of Mathematical Sciences and Applications

Volume 6 (2021)

Number 2

A structure-preserving method for solving the complex $\mathsf{T}$-Hamiltonian eigenvalue problem

Pages: 199 – 224



Heng Tian (Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan)

Xing-Long Lyu (School of Mathematics, Southeast University, Nanjing, China)

Tiexiang Li (School of Mathematics, Southeast University, Nanjing, China; and Nanjing Center for Applied Mathematics, Nanjing, China)


In this work, we present a new structure-preserving method to compute the structured Schur form of a dense complex $\mathsf{T}$ Hamiltonian matrix $\mathscr{H}$ of moderate size. Origination of the complex $\mathsf{T}$ Hamiltonian eigenvalue problem outside the control theory is briefly discussed. Specifically, our method consists of three main stages. At the first stage, we compute eigenvalues of $\mathscr{H}$ using the $\mathsf{T}$ symplectic URV-decomposition of complex $\mathscr{H}$ followed up with the complex periodic QR algorithm to thoroughly respect the $(\lambda,-\lambda)$ pairing of eigenvalues. At the second stage, we construct the $\mathsf{T}$ isotropic invariance subspace of $\mathscr{H}$ from suitable linear combination of columns of $U$ and $V$ matrices from the first stage. At the third stage, we find a $\mathsf{T}$ symplectic-orthogonal basis of this invariance subspace, which immediately provides the structured Schur form of $\mathscr{H}$. Several numerical results are presented to demonstrate the effectiveness and accuracy of our method.


complex $\mathsf{T}$ Hamiltonian eigenvalue problem, $\mathsf{T}$ symplectic URV-decomposition, complex periodic QR algorithm, complex $\mathsf{T}$ Hamiltonian Schur form

H. Tian was supported by the Ministry of Science and Technology (MoST) 107-2811-M-009-002. T. Li was supported by the National Natural Science Foundation of China (NSFC) 11971105. This work was also partially supported by the ST Yau Centre in Taiwan, Shing-Tung Yau Center and Big Data Computing Center of Southeast University.

Received 1 April 2021

Accepted 11 May 2021

Published 18 October 2021