Open AccessOn the Existence of Schur-like Forms for Matrices with Symmetry Structures
Author(s) -
Christian Mehl
Publication year - 2020
Publication title -
vietnam journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.443
H-Index - 13
eISSN - 2305-2228
pISSN - 2305-221X
DOI - 10.1007/s10013-020-00394-3
Subject(s) - mathematics , schur complement , unitary state , pure mathematics , hermitian matrix , schur decomposition , unitary matrix , eigenvalues and eigenvectors , hamiltonian (control theory) , schur product theorem , matrix similarity , hamiltonian matrix , unitary transformation , schur's theorem , matrix (chemical analysis) , circular ensemble , algebra over a field , hadamard product , symmetric matrix , mathematical analysis , quantum , quantum mechanics , partial differential equation , orthogonal polynomials , materials science , law , gegenbauer polynomials , composite material , classical orthogonal polynomials , mathematical optimization , political science , physics , hadamard transform
Schur-like forms are developed for matrices that have a symmetry structure with respect to an indefinite inner product induced by a Hermitian and unitary Gram matrix. It is characterized under which conditions these forms can be computed by structure-preserving unitary transformations. The main results combines and generalizes the two well-known results from the literature that on the one hand any normal matrix can be unitarily diagonalized and on the other hand a Hamiltonian matrix can be transformed to Hamiltonian Schur form via a unitary similarity transformation if and only if its purely imaginary eigenvalues satisfy certain conditions that involve the sign characteristic of the matrix under consideration.