Coherent algebra

A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix formula_1 and the all-ones matrix formula_2.
Definitions.
A subspace formula_3 of formula_4 is said to be a coherent algebra of order formula_5 if:
A coherent algebra formula_12 is said to be:
The set formula_16 of "Schur-primitive matrices" in a coherent algebra formula_3 is defined as formula_18.
Dually, the set formula_19 of "primitive matrices" in a coherent algebra formula_3 is defined as formula_21.