Norm-preserving dilation theorems for a block positive semidefinite (definite) matrix. Issue 22 (21st December 2022)

• Record Type: Journal Article
• Authors: Liu, Xifu
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 22(2022)
• Page Start: 7770
• Page End: 7777
• Publication Date: 2022-12-21
• Subjects: Norm-preserving dilation -- spectral norm -- positive semidefinite matrix -- positive definite matrix -- block matrix; 15A09 -- 15A24 -- 15A60 -- 47A20; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2021.2008854
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : Consider the following problem: given a positive semidefinite (definite) matrix A ∈ C m × m, and B ∈ C n × m, such that ‖ ( A B ) ‖ 2 = μ, find a matrix X satisfying 1 ‖ ( A B ∗ B X ) ‖ 2 = μ subject to ( A B ∗ B X ) ⩾ ( > ) 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.