More results related to geometric mean and singular values for matrices. (English) Zbl 1486.15023
Summary: In 2016, M. Lin [Electron. J. Linear Algebra 31, 120–124 (2016; Zbl 1341.15015)]
conjectured that if \( \begin{pmatrix} A & B \\ B^* & C \end{pmatrix} \in \mathbb M_2(\mathbb M_n)\) is a positive semi-definite matrix then \(s_j(\Phi(B)) \leq s_j(\Phi(A) \sharp \Phi(C))\), \( j=1,2,\ldots,n\) where \(\Phi(X) = X + Tr(X) I_n\) and \(s_j(.)\) means the \(j\) th largest singular value. In this note, we confirm this conjecture when \(AB = BA\) and prove the more general result
\[
s_j(\Psi_f(B)) \leq s_j( \Psi_f(A) \sharp \Psi_f(C)) \quad \text{and}\quad s_j(\Psi_f(|B|)) \leq s_j( \Psi_f(A \sharp C)),\;\;\; j = 1,2,\ldots,n
\]
where \(\Psi(X) = X + f(X) I_n\) and \(f\) is a Liebian function. Some related inequalities are also investigated.
MSC:
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
Keywords:
positive semi-definite matrices; geometric mean; singular values inequalities for matrices; Liebian functionsCitations:
Zbl 1341.15015References:
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