Symmetry properties of and reduction principles for divisibility relations between the invariant factors of products of holomorphic matrix functions Original Research Article

The invariant factors λγnλγn − 1 … λγ2λγ1 of a product C = AB of n × n matrices which are analytic in a neighbourhood of 0 with invertible values for λ ≠ 0 are related to the invariant factors λαnλαn − 1 … λα2λα1 (λβnλβn − 1 … λβ2λβ1) of A (B) through divisibility relations of the type imageγr1 + γr2 + cdots, three dots, centered + γrk ≤ αs1 + αs2 + cdots, three dots, centered + αsk + βt1 +βt2 +cdots, three dots, centered+ βtk for certain (ordered) index sets r1 < r2 < … < rk ≤ n, s1 < s2 < … < sk less-than-or-equals, slant n, t1 < t2 < … < tk less-than-or-equals, slant n. The index sets for which such divisibility relations hold display certain symmetries, allowing us to relate one such index set to up to eleven index sets for which the divisibility relation is also valid. Further, in many cases the proof of the divisibility relation can be reduced to the proof of a similar relation involving less indices or matrices of a lower order. Both these symmetry results and the reduction techniques are described and used for the derivation of several systems of divisibility relations, including the complete description for the cases where k ≤ 3 and k ≥ n − 3.