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Inequalities for certain powers of positive definite matrices. (English) Zbl 1426.15028
Summary: Let \(A,B\), and \(X\) be \(n \times n\) matrices such that \(A,B\) are positive definite and \(X\) is Hermitian. If \(a\) and \(b\) are real numbers such that \(0< a \leqslant s_n(A)\) and \(0 < b \leqslant s_n(B)\), then it is shown, among other inequalities, that \[|||A^bX+ XB^a||| \geqslant (1 + \min(a^2,b^2))|||X|||\] for every unitarily unitarily invariant norm.
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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