A simultaneous decomposition of a matrix triplet with applications

作者：Liu, Yonghui^*; Tian, Yongge

来源：Numerical Linear Algebra with Applications, 2011, 18(1): 69-85.

DOI：10.1002/nla.701

• singular value decomposition
• eigenvalue decomposition
• simultaneous matrix decomposition
• matrix expressions
• matrix equation
• maximal rank
• minimal rank

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A = +/-A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A-BXB*-CYC* and then solve two conjectures on the maximal and minimal possible ranks of A-BXB*-CYC* with respect to X = +/-X* and Y = +/-Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC = A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation.

• 出版日期2011-1
• 单位中央财经大学; 上海金融学院