Further relationships between certain partial orders of matrices and their squares
J. K. Baksalary1 , O. M. Baksalary2 , X. Liu3,4
1 Institute of Mathematics, Zielona Góra University, ul. Podgórna 50, PL 65-246, Zielona Góra, Poland
2 Institute of Physics, Adam Mickiewicz University, ul. Umultowska 85, PL 61-614, Poznań , Poland
3 Department of Applied Mathematics, Xidian University, Xian 710071, People’s Republic of China
4 Department of Mathematics, Suzhou Railway Teacher’s College, Suzhou 215009, People’s Republic of China
Linear Algebra and its Applications, 375, 171-180 (2003)
Theorem 3 of Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] asserts that if both A and B are Hermitian nonnegative definite matrices, then the star order A<=*B between them and the star order A2<=*B2 between their squares are equivalent and they imply the commutativity property AB=BA. In this paper, relationships between the three conditions mentioned above are reinvestigated in situations where the assumptions on A and B are completely or partially relaxed. Some results concerning the star order are obtained as corollaries to corresponding results referring to the left-star and right-star orders introduced by Baksalary and Mitra [Linear Algebra Appl. 149 (1991) 73].