Norm-preserving dilation theorems for a block positive semidefinite (definite) matrix
Xifu Liu · Linear and Multilinear Algebra · 2021
Consider the following problem: given a positive semidefinite (definite) matrix A∈Cm×m, and B∈Cn×m, such that ‖(AB)‖2=μ, find a matrix X satisfying ‖(AB∗BX)‖2=μsubjectto(AB∗BX)⩾(>)0.In 1996, Zheng presented the expressions for X formed by Löwner partial ordering when the problem is solvable, and pointed out that there may be no X satisfying these expressions. In this paper, we present some sufficient and necessary conditions for the existence of X and provide its explicit general expressions.