A treatment of a determinant inequality of Fiedler and Markham
Minghua Lin · Czechoslovak Mathematical Journal · 2016
Fiedler and Markham (1994) proved $${\left( {\frac{{\det \hat H}}{k}} \right)^k} \geqslant \det H,$$ where H = (H ij ) i,j=1 n is a positive semidefinite matrix partitioned into n × n blocks with each block k × k and $$\hat H = \left( {tr{H_{ij}}} \right)_{i,j = 1}^n$$ . We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $$\det \left( {{I_n} + \hat H} \right) \geqslant \det {\left( {{I_{nk}} + kH} \right)^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern- ulldelimiterspace} k}}}.$$