Inequalities for Contraction Matrices. (11th June 2019) - Abu-As'ad, Ata; Hirzallah, Omar
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.