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A treatment of a determinant inequality of Fiedler and Markham. (English) Zbl 1413.15040

Summary: M. Fiedler and T. L. Markham [Math. Slovaca 44, No. 4, 441–444 (1994; Zbl 0828.15023)] proved \[ \Big(\frac{\det\widehat{H}}{k}\Big)^{k}\geq\det H, \] where \(H=(H_{ij})_{i,j=1}^n\) is a positive semidefinite matrix partitioned into \(n\times n\) blocks with each block \(k\times k\) and \(\widehat{H}=(\operatorname{tr}H_{ij})_{i,j=1}^n\). We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove \[ \det(I_n+\widehat{H})\geq\det(I_{nk}+kH)^{1/k}. \]

MSC:

15A45 Miscellaneous inequalities involving matrices
15A15 Determinants, permanents, traces, other special matrix functions

Citations:

Zbl 0828.15023
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References:

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