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Inequalities for contraction matrices. (English) Zbl 1411.15010
Summary: Let \(A\), \(B\), \(C\), \(X\) and \(Y\) be \(n\times n\) matrices such that \(A\) and \(B\) are positive definite contractions. It is shown that if \( r\geq s_n(A)\) and \(t\geq s_n(B)\) then \[\| A^{-r} X+XB^{-t}\|^2_2 + \| AX+XB\|^2_2 \leq 4 \| AXB^{-1}+A^{-1} XB\|^2_2.\] Moreover, if \(0<Y\leq X\leq C+Y\leq 2C\), then \[s_j\biggl( (C+X)^{-1/2} A(C+Y)^{-1/2}\biggr) \leq \frac{\kappa (C)}{\|C\|+\sqrt{s_{n-j+i}(X)_{n-j+i}(Y)}}\] for \(i\), \(j=1,\dots,n\) with \(i\leq j\leq 2i-1\) where \(\|T\|_2\), \(\|T\|\), \(s_j (T)\) and \(\kappa (T)\) denote the Hilbert-Schmidt norm, the spectral matrix norm, the \(j\) th singular value, and the condition number of the \(n\times n\) matrix \(T\), respectively.

15A45 Miscellaneous inequalities involving matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
Full Text: DOI
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