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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.
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
[1] A. ABU-AS’AD ANDO. HIRZALLAH, Some inequalities for powers of positive definite matrices, Math. Ineq. Appl. 20 (2017), 13-27. · Zbl 1362.15016
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