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Positive matrices partitioned into a small number of Hermitian blocks. (English) Zbl 1262.15037

The properties of positive matrices partitioned into blocks are studied. The geometric mean of two \(n \times n\) matrices \(A, B \in {\mathbb M}_n^+\) (the space of positive semidefinite matrices) is defined as the largest possible Hermitian matrix \(X\) such that the matrix \[ H = \left[ \begin{matrix} A & X \\ X & B \end{matrix} \right] \] in \({\mathbb M}_{2n}^+\) is positive. All symmetric norms of the positive matrices of the class of such block matrices \(H\) are characterized by an inequality \(\|H\| \leq \|A + B\|\). Generally, for \( H= [A_{s, t}] \in {\mathbb M}_{\alpha n}^+\) partitioned into \(\alpha \times \alpha \) Hermitian blocks in \({\mathbb M}_n^+\) the inequality \(\|H\| \leq \|\Delta\|\) is known, where \(\Delta = \sum^{\alpha}_{s = 1} A_{s, s}\) is the so-called partial trace. This inequality is further analyzed for special cases of small partitions \(\alpha \in \{ 2, 3, 4 \}\). It is shown in these special cases that proofs provide sharper decomposition than in the general case and that for a positive matrix with real entries the decomposition involves some complex matrices. The analysis for the case with \(\alpha = 4\) leads to introducing the quaternions.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B57 Hermitian, skew-Hermitian, and related matrices
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A45 Miscellaneous inequalities involving matrices
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