Inequalities for Contraction Matrices. (11th June 2019)

• Record Type: Journal Article
• Authors: Abu-As'ad, Ata; Hirzallah, Omar
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 8(2019)
• Page Start: 980
• Page End: 991
• Publication Date: 2019-06-11
• Subjects: A contraction matrix -- condition number of a matrix -- Hilbert–Schmidt norm -- norm inequality -- positive definite matrix -- singular value -- spectral norm; 15A18 -- 15A42 -- 15A45 -- 15A60; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1596952
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: LetA, B, C, X, and Y be n × n matrices such that A and B are positive definite contractions. It is shown that ifr ≥ s n ( A ) andt ≥ s n ( B ), then‖ A − r X + X B − t ‖ 2 2 + ‖ AX + XB ‖ 2 2 ≤ 4 ‖ AX B − 1 + A − 1 X B ‖ 2 2 . Moreover, if0 < Y ≤ X ≤ C + Y ≤ 2 C, thens j ( ( C + X ) − 1 / 2 A ( C + Y ) − 1 / 2 ) ≤ κ ( C ) ‖ C ‖ + s n − j + i ( X ) s n − j + i ( Y ) fori, j = 1, …, n withi ≤ j ≤ 2 i − 1, where‖ T ‖ 2, ‖ T ‖, s j ( T ), andκ ( T ) denote the Hilbert-Schmidt norm, the spectral matrix norm, the j th singular value, and the condition number of the n × n matrix T, respectively.