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### Squaring a reverse AM-GM inequality

Studia Mathematica

Let A, B be positive operators on a Hilbert space with 0 < m ≤ A, B ≤ M. Then for every unital positive linear map Φ, Φ²((A + B)/2) ≤ K²(h)Φ²(A ♯ B), and Φ²((A+B)/2) ≤ K²(h)(Φ(A) ♯ Φ(B))², where A ♯ B is the geometric mean and K(h) = (h+1)²/(4h) with h = M/m.

### A treatment of a determinant inequality of Fiedler and Markham

Czechoslovak Mathematical Journal

Fiedler and Markham (1994) proved ${\left(\frac{\mathrm{det}\stackrel{^}{H}}{k}\right)}^{k}\ge \mathrm{det}H,$ where $H={\left({H}_{ij}\right)}_{i,j=1}^{n}$ is a positive semidefinite matrix partitioned into $n×n$ blocks with each block $k×k$ and $\stackrel{^}{H}={\left(\mathrm{tr}{H}_{ij}\right)}_{i,j=1}^{n}$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $\mathrm{det}\left({I}_{n}+\stackrel{^}{H}\right)\ge \mathrm{det}{\left({I}_{nk}+kH\right)}^{1/k}.$

### Bilinear characterizations of companion matrices

Special Matrices

Companion matrices of the second type are characterized by properties that involve bilinear maps.

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