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Title: Inequalities for Contraction Matrices
Other Titles: Matrix analysis
Authors: Abu-Asa'd, Ata
Keywords: A contraction matrix, condition number of a matrix, Hilbert–Schmidt norm, norm inequality, positive definite matrix,
singular value, spectral norm
Issue Date: 4-Apr-2019
Publisher: Numerical Functional Analysis and Optimization
Series/Report no.: Volume 40, 2019 - Issue 8;Pages 980-991
Abstract: Let A,B,C,X, and Y be n× n matrices such that A and B are positive definite contractions. It is shown that if r≥sn(A) and t≥sn(B), then ∥A−rX+XB−t∥22+∥AX+XB∥22≤4∥∥AXB−1+A−1XB∥∥22. Moreover, if 0<Y≤X≤C+Y≤2C, then sj((C+X)−1/2A(C+Y)−1/2)≤κ(C)∥C∥+sn−j+i(X)sn−j+i(Y)−−−−−−−−−−−−−−−√ for i,j=1,…,n with i≤j≤2i−1, where ∥T∥2,∥T∥,sj(T), and κ(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.
Description: Let MnðCÞ be the algebra of all n n complex matrices. The singular values s1ðAÞ; :::; snðAÞ of a matrix A 2 MnðCÞ are the eigenvalues of the matrix ðA AÞ1=2 arranged in decreasing order and repeated according to multiplicity. A Hermitian matrix A 2 MnðCÞ is said to be positive semidefinite, written as A 0; if x Ax 0 for all x 2 Cn and it is called positive definite, written as A>0, if x Ax>0 for all x 2 Cn with x 6¼ 0: The Hilbert–Schmidt norm (or the Frobenius norm) k k2 is the norm defined on MnðCÞ by kAk2 ¼ ð Pnj ¼1 s2j ðAÞÞ1=2;A 2 MnðCÞ: The Hilbert–Schmidt norm is unitarily invariant, that is kUAVk2 ¼ kAk2 for all A 2 MnðCÞ and all unitary matrices U; V 2 MnðCÞ: Another property of the Hilbert–Schmidt norm is that kAk2 ¼ ð Pn i;j¼1 jf j Aeij2Þ1=2; where fejgnj ¼1 and ffjgnj ¼1 are two orthonormal bases of Cn: The spectral matrix norm, denoted by k k; of a matrix A 2 MnðCÞ is the norm defined by kAk ¼ supfkAxk : x 2 Cn; kxk ¼ 1g or equivalently kAk ¼ s1ðAÞ; For further properties of these norms,
ISSN: 0163-0563 (Print) 1532-2467 (online)
Appears in Collections:Applied science faculty

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