## Abstract

Various trace minimization principles have interplayed with numerical computations for the standard eigenvalue and generalized eigenvalue problems in general, as well as important applied eigenvalue problems including the linear response eigenvalue problem from electronic structure calculation and the symplectic eigenvalue problem of positive definite matrices that play important roles in classical Hamiltonian dynamics, quantum mechanics, and quantum information, among others. In this paper, Ky Fan's trace minimization principle is extended along the line of the Brockett cost function tr(DX^{H}AX) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we investigate inf_{X}tr(DX^{H}AX) subject to X^{H}BX=I_{k} (the k×k identity matrix) or X^{H}BX=J_{k}, where J_{k}=diag(±1). We establish conditions under which the infimum is finite and when it is finite, analytic solutions are obtained in terms of the eigenvalues and eigenvectors of the matrix pencil A−λB, where B is possibly indefinite and possibly singular, and D is also possibly indefinite.

Original language | English |
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Pages (from-to) | 483-509 |

Number of pages | 27 |

Journal | Linear Algebra and Its Applications |

Volume | 656 |

Early online date | 13 Oct 2022 |

DOIs | |

Publication status | Published - 1 Jan 2023 |

## Scopus Subject Areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

## User-Defined Keywords

- Brockett cost function
- Eigenvalue
- Eigenvector
- Linear response eigenvalue problem
- Symplectic eigenvalue problem of positive definite matrix
- Trace minimization principle