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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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