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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.

MSC:
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
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