Further inequalities for certain powers of positive definite matrices. (English) Zbl 1512.15021
Summary: Let \(A_i\), \(P\), \(i = 1, \dots, m\), be \(n\) by \(n\) complex matrices such that each \(A_i\) is positive definite. It is shown, among other inequalities, that
- (a)
- If \(0 < a_j \leq s_n(A_j)\), \(C_m = \sum^m_{j=1}(\sum^m_{i=1, i \neq j} A_i)^{a_j}\) and \(P\) is Hermitian, then for every unitarily invariant norm \(\| |.|\| \), \[ \| |C_m P + PC_m\| | \geq \alpha\| |P|\|, \] where \(\alpha = \max \{2 (m - 1), m (1 + \min \{a^2_1, \dots, a^2_m\})\}\).
- (b)
- If \(s_1(A_j) \leq b_j\), \(\sum^m_{i=1} b_i = 1\) and \(D_m = \sum^m_{j=1}(\sum^m_{i=1, i \neq j} A_i)^{b_j}\), then for every unitarily invariant norm \(\| |.|\|\), \[ 2m \| |P|\| \geq \| |D_m P + PD_m|\|. \]
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 |
| 26A51 | Convexity of real functions in one variable, generalizations |
Keywords:
positive definite matrix; unitarily invariant norm; singular value; convex function; inequalityReferences:
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