Positive matrices partitioned into a small number of Hermitian blocks

Positive semidefinite matrices partitioned into a small number of Hermitian blocks have a remarkable property. Such a matrix may be written in a simple way from the sum of its diagonal blocks: the full matrix is a kind of average of copies of the sum of the diagonal blocks. This entails several eigenvalue inequalities. The proofs use a decomposition lemma for positive matrices, isometries with complex entries, and the Pauli matrices.