## 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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