Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

Abstract

For any A=A ₁+A ₂ j∈Q ^n×n and η∈<texlscub>i, j, k</texlscub>, denote A ^{η H} =−η A ^H η. If A ^{η H} =A, A is called an $\eta$-Hermitian matrix. If A ^{η H} =−A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ ^n×n and η AQ ^n×n , respectively.

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equation AXB+CYD=E over X∈η HQ ^n×n and Y∈η AQ ^n×n .

Keywords:

• matrix equation
• least-squares solution
• Moore–Penrose generalized inverse
• Kronecker product
• η-Hermitian matrices

2010 AMS Subject Classifications::

• 65F05
• 65H10
• 15B33

Acknowledgements

The authors thank the referees very much for their valuable suggestions and comments, which resulted in a great improvement of the original manuscript. This work is supported by the Natural Science Foundation of China (11171205, 60672160), Natural Science Foundation of Shanghai (11ZR1412500), PhD Programs Foundation of Ministry of Education of China (20093108110001), the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), Shanghai Leading Academic Discipline Project (J50101), Guangdong Natural Science Fund of China (No. 10452902001005845), and Program for Guangdong Excellent Talents in University, Guangdong Education Ministry, China (LYM10128), Science and Technology Project of Jiangmen City, China (No. 201215628).