Let $ J = \left[ \begin{array}{cc} 0 & I_n \\ -I_n & 0 \\ \end{array} \right] \in \mathbb{R}^{2n\times 2n} $. A matrix $ A \in \mathbb{R}^{2n\times 2n} $ is said to be Hamiltonian if $ (AJ)^{\top} = AJ $. In this paper, we first consider the following generalized inverse eigenvalue problem (GIEP): Given a pair of matrices $ (\Lambda, X) $ in the form $ \Lambda = {\rm{diag}}\{\lambda_1, \cdots, \lambda_{p}\}\in \mathbb{C}^{p\times p} $ and $ X = [{\bf{x}}_{1}, \cdots, {\bf{x}}_{p}]\in \mathbb{C}^{2n\times p} $, where diagonal elements of $ \Lambda $ are all distinct with rank$ (X) = p $, and both $ \Lambda $ and $ X $ are closed under complex conjugation in the sense that $ \lambda_{2i} = \bar{\lambda}_{2i-1}\in \mathbb{C}, $ $ {\bf{x}}_{2i} = \bar{{\bf{x}}}_{2i-1}\in \mathbb{C}^{2n} $ for $ i = 1, \cdots, l, $ and $ \lambda_{j}\in \mathbb{R}, $ $ {\bf{x}}_{j}\in \mathbb{R}^{2n} $ for $ j = 2l+1, \cdots, p. $ Find Hamiltonian matrices $ A $ and $ B $ such that $ AX\Lambda = BX. $ Then, we consider the associated optimal approximation problem (OAP): Given $ \tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n} $. Find $ (\hat{A}, \hat{B}) \in \mathbb{S_{E}} $ such that $ \|\hat{A}-\tilde{A}\|^2+\|\hat{B}-\tilde{B}\|^2 = {\min_{(A, B)\in \mathbb{S_{E}}}}\left(\|A-\tilde{A}\|^2+\|B-\tilde{B}\|^2\right), $ where $ \mathbb{S_{E}} $ is the solution set of Problem GIEP. By using the QR-decomposition, we deduce the representation of the general solution of Problem GIEP. Also, we obtain the unique optimal approximation solution $ (\hat{A}, \hat{B}) $ of Problem OAP.
Citation: Lina Liu, Huiting Zhang, Yinlan Chen. The generalized inverse eigenvalue problem of Hamiltonian matrices and its approximation[J]. AIMS Mathematics, 2021, 6(9): 9886-9898. doi: 10.3934/math.2021574
Let $ J = \left[ \begin{array}{cc} 0 & I_n \\ -I_n & 0 \\ \end{array} \right] \in \mathbb{R}^{2n\times 2n} $. A matrix $ A \in \mathbb{R}^{2n\times 2n} $ is said to be Hamiltonian if $ (AJ)^{\top} = AJ $. In this paper, we first consider the following generalized inverse eigenvalue problem (GIEP): Given a pair of matrices $ (\Lambda, X) $ in the form $ \Lambda = {\rm{diag}}\{\lambda_1, \cdots, \lambda_{p}\}\in \mathbb{C}^{p\times p} $ and $ X = [{\bf{x}}_{1}, \cdots, {\bf{x}}_{p}]\in \mathbb{C}^{2n\times p} $, where diagonal elements of $ \Lambda $ are all distinct with rank$ (X) = p $, and both $ \Lambda $ and $ X $ are closed under complex conjugation in the sense that $ \lambda_{2i} = \bar{\lambda}_{2i-1}\in \mathbb{C}, $ $ {\bf{x}}_{2i} = \bar{{\bf{x}}}_{2i-1}\in \mathbb{C}^{2n} $ for $ i = 1, \cdots, l, $ and $ \lambda_{j}\in \mathbb{R}, $ $ {\bf{x}}_{j}\in \mathbb{R}^{2n} $ for $ j = 2l+1, \cdots, p. $ Find Hamiltonian matrices $ A $ and $ B $ such that $ AX\Lambda = BX. $ Then, we consider the associated optimal approximation problem (OAP): Given $ \tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n} $. Find $ (\hat{A}, \hat{B}) \in \mathbb{S_{E}} $ such that $ \|\hat{A}-\tilde{A}\|^2+\|\hat{B}-\tilde{B}\|^2 = {\min_{(A, B)\in \mathbb{S_{E}}}}\left(\|A-\tilde{A}\|^2+\|B-\tilde{B}\|^2\right), $ where $ \mathbb{S_{E}} $ is the solution set of Problem GIEP. By using the QR-decomposition, we deduce the representation of the general solution of Problem GIEP. Also, we obtain the unique optimal approximation solution $ (\hat{A}, \hat{B}) $ of Problem OAP.
| [1] | K. R. Meyer, G. R. Hall, Introduction to Hamiltonian dynamical systems and the N-body problem, Spring, New York, 1992. |
| [2] |
B. J. Leimkuhler, E. S. V. Vleck, Orthosymplectic integration of linear Hamiltonian systems, Numer. Math., 77 (1997), 269–282. doi: 10.1007/s002110050286
|
| [3] | V. Mehrmann, The autonomous linear quadratic control problem: Theory and numerical solution, Springer-Verlag, 1991. |
| [4] | K. Zhou, J. C. Doyle, K. Glover, Robust and optimal control, Prentice Hall, New Jersey, 1996. |
| [5] |
I. L. Thomas, Hamiltonian matrix elements from a symmetric wave function, Phys. Rev. A, 2 (1970), 728–733. doi: 10.1103/PhysRevA.2.728
|
| [6] |
P. Mullen, Y. Tong, P. Alliez, M. Desbrun, Spectral conformal parameterization, Comput. Graph. Forum, 27 (2008), 1487–1494. doi: 10.1111/j.1467-8659.2008.01289.x
|
| [7] |
W. Q. Huang, X. D. Gu, W. W. Lin, S. T. Yau, A novel symmetric skew-Hamiltonian isotropic Lanczos algorithm for spectral conformal parameterizations, J. Sci. Comput., 61 (2014), 558–583. doi: 10.1007/s10915-014-9840-2
|
| [8] |
J. J. Dongarra, J. R. Gabriel, D. D. Koelling, J. H. Wilkinson, The eigenvalue problem for Hermitian matrices with time-reversal symmetry, Linear Algebra Appl., 60 (1984), 27–42. doi: 10.1016/0024-3795(84)90068-5
|
| [9] |
N. Rösch, Time-reversal symmetry, Kramers' degeneracy and the algebraic eigenvalue problem, Chem. Phys., 80 (1983), 1–5. doi: 10.1016/0301-0104(83)85163-5
|
| [10] |
J. Olsen, P. JØrgensen, Linear and nonlinear response functions for an exact state and for an MCSCF state, J. Chem. Phys., 82 (1985), 3235–3264. doi: 10.1063/1.448223
|
| [11] |
J. Olsen, H. JØrgen, A. Jensen, P. JØrgensen, Solution of the large matrix equations which occur in response theory, J. Comput. Phys., 74 (1988), 265–282. doi: 10.1016/0021-9991(88)90081-2
|
| [12] | P. Lancaster, L. Rodman, Algebraic Riccati equations, Clarendon press, 1995. |
| [13] | Z. Zhang, X. Hu, L. Zhang, The solvability conditions for the inverse eigenvalue problem of Hermitian-generalized skew-Hamiltonian matrices and its approximation, Inverse Probl., 18 (2002) 1369–1376. |
| [14] |
Z. Bai, The solvability conditions for the inverse eigenvalue problem of Hermitian and generalized skew-Hamiltonian matrices and its approximation, Inverse Probl., 19 (2003), 1185–1194. doi: 10.1088/0266-5611/19/5/310
|
| [15] |
J. Qian, R. C. E. Tan, On some inverse eigenvalue problems for Hermitian and generalized Hamiltonian/skew-Hamiltonian matrices, J. Comput. Appl. Math., 250 (2013), 28–38. doi: 10.1016/j.cam.2013.02.023
|
| [16] |
S. Gigola, L. Lebtahi, N. Thome, Inverse eigenvalue problem for normal $J$-hamiltonian matrices, Appl. Math. Lett., 48 (2015), 36–40. doi: 10.1016/j.aml.2015.03.007
|
| [17] |
S. Gigola, L. Lebtahi, N. Thome, The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem, J. Comput. Appl. Math., 291 (2016), 449–457. doi: 10.1016/j.cam.2015.03.052
|
| [18] |
K. T. Joseph, Inverse eigenvalue problem in structral design, AIAA J., 30 (1992), 2890–2896. doi: 10.2514/3.11634
|
| [19] | S. Li, B. Wang, J. Hu, Homotopy solution of the inverse generalized eigenvalue problems in structural dynamics, Appl. Math. Mech. 25 (2004), 580–586. |
| [20] | Y. Yuan, A symmetric inverse eigenvalue problem in structural dynamic model updating, Appl. Math. Comput., 213 (2009), 516–521. |
| [21] |
Y. Yuan, H. Dai, A generalized inverse eigenvalue problem in structural dynamic model updating, J. Comput. Appl. Math., 226 (2009), 42–49. doi: 10.1016/j.cam.2008.05.015
|
| [22] |
P. Wang, H. Dai, Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse, J. Eng. Math., 96 (2016), 201–210. doi: 10.1007/s10665-015-9790-1
|
| [23] |
K. V. Singh, H. Ouyang, Pole assignment using state feedback with time delay in friction-induced vibration problems, Acta Mech., 224 (2013), 645–656. doi: 10.1007/s00707-012-0778-x
|
| [24] |
J. Zhao, J. Zhang, The solvability conditions for the inverse eigenvalue problem of normal skew $J$-Hamiltonian matrices, J. Inequal. Appl., 2018 (2018), 1–8. doi: 10.1186/s13660-017-1594-6
|
| [25] | H. Zhang, Y. Yuan, Generalized inverse eigenvalue problems for Hermitian and $J$-Hamiltonian/skew-Hamiltonian matrices, Appl. Math. Comput., 361 (2019), 609–616. |
| [26] |
Y. Yuan, J. Chen, An inverse eigenvalue problem for Hamiltonian matrices, J. Comput. Appl. Math., 381 (2021), 113031. doi: 10.1016/j.cam.2020.113031
|
| [27] | J. P. Aubin, Applied functional analysis, John Wiley & Sons, New York, 1979. |
Lina Liu, Huiting Zhang, Yinlan Chen. The generalized inverse eigenvalue problem of Hamiltonian matrices and its approximation[J]. AIMS Mathematics, 2021, 6(9): 9886-9898. doi: 10.3934/math.2021574
DownLoad: