Simultaneous decomposition of quaternion matrices involving eta-Hermicity with applications

作者：He, Zhuo-Heng; Wang, Qing-Wen^*; Zhang, Yang

来源：Applied Mathematics and Computation, 2017, 298: 13-35.

DOI：10.1016/j.amc.2016.11.002

• The quaternion algebra
• Matrix equations
• Matrix decompositions
• eta-Hermitian matrix
• Rank

Let R and (HXn)-X-m stand, respectively, for the real number field and the set of all m X n matrices over the real quaternion algebra
H = {a(0) + a(1)i + a(2)j + a(3)k vertical bar i(2) = j(2) = k(2) = ijk = -1 , a(0) , a(1) , a(2) , a(3) epsilon R}.
For eta epsilon {i, j, k}, a real quaternion matrix A epsilon (HXn)-X-n is said to be eta-Hermitian if A(eta)* = A where A(eta)* = - eta A*eta , and A* stands for the conjugate transpose of A, arising in widely linear modeling. We present a simultaneous decomposition for a set of nine real quaternion matrices involving eta-Hermicity with compatible sizes: A(i) epsilon (HXti)-X-pi, B-i epsilon (HXti+1)-X-pi , and C-i epsilon (HXpi)-X-pi , where C-i are eta-Hermitian matrices, ( i = 1 , 2 , 3 ) . As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an eta-Hermitian solution to the system of coupled real quaternion matrix equations A(i)X(i)A(i)(eta)* + BiXi+1 B-i(eta)* = C-i , ( i = 1 , 2 , 3 ) , and provide an expression of the general eta-Hermitian solutions to this system. Moreover, we derive the rank bounds of the general eta-Hermitian solutions to the above-mentioned system using ranks of the given matrices A(i), B-i, and C-i as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper.

• 出版日期2017-4-1
• 单位上海大学