Linear maps transforming H-unitary matrices

Chi Kwong Li, Nung Sing Sze

Research output: Journal article publicationJournal articleAcademic researchpeer-review

2 Citations (Scopus)

Abstract

Let H1be an n × n invertible Hermitian matrix, and let U(H1) be the group of n × n H1-unitary matrices, i.e., matrices A satisfying A*H1A = H1. Suppose H2is an m × m invertible Hermitian matrix. We show that a linear transformation φ: Mn→ Mmsatisfies φ(U(H1))⊂U(H2) if and only if there exist invertible matrices S ∈ Mm, U,V ∈ U(H2) such that S* H2S = [(Ia⊕ - Ib) ⊗ H1] ⊕[(Ic⊕-Id) ⊗ (H1-1)t],and φ has the form A → US[(Ia+b⊗ A) ⊕ (Ic+d⊗ At)]S-1V, where a, b, c and d are nonnegative integers satisfying (a+b+c+d)n = m. Assume H1has inertia (p,q) and H2has inertia (r,s). Then there is a linear transformation mapping U(H1) into U(H2) if and only if there are nonnegative integers u and v such that (r, s) = u(p,q)+v(q,p). These results generalize those of Marcus, Cheung and Li.
Original languageEnglish
Pages (from-to)111-124
Number of pages14
JournalLinear Algebra and Its Applications
Volume377
Issue number1-3
DOIs
Publication statusPublished - 15 Jan 2004
Externally publishedYes

Keywords

  • H-unitary matrices
  • Indefinite inner product
  • Linear transformations

ASJC Scopus subject areas

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

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