Open AccessSome quaternion matrix equations involving Φ-Hermicity
Author(s) -
Zhuo-Heng He
Publication year - 2019
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1916097h
Subject(s) - mathematics , quaternion , hermitian matrix , quaternion algebra , matrix (chemical analysis) , algebra over a field , pure mathematics , geometry , algebra representation , division algebra , materials science , composite material
Let H be the real quaternion algebra and Hmxn denote the set of all m x n matrices over H. For A ? Hm x n, we denote by A? the n x m matrix obtained by applying ? entrywise to the transposed matrix At, where ? is a nonstandard involution of H. A ? Hnxn is said to be ?-Hermitian if A = A?. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number (A,B,C,D), where A is ?-Hermitian, and B,C,D are general matrices. Using this simultaneous matrix decomposition, we derive necessary and sufficient conditions for the existence of a solution to some real quaternion matrix equations involving ?-Hermicity in terms of ranks of the given real quaternion matrices. We also present the general solutions to these real quaternion matrix equations when they are solvable. Finally some numerical examples are presented to illustrate the results of this paper.