Article

Integral Equations and Operator Theory

, Volume 16, Issue 2, pp 277-304

First online:

The local invariant factors of a product of holomorphic matrix functions: The order 4 case

  • G. PhilipAffiliated withEconometrisch Instituut, Erasmus Universiteit
  • , A. ThijsseAffiliated withEconometrisch Instituut, Erasmus Universiteit

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Abstract

Let\(\lambda ^{\alpha _n } |\lambda ^{\alpha _{n - 1} } |...|\lambda ^{\alpha _2 } |\lambda ^{\alpha _1 } \), resp.\(\lambda ^{\beta _n } |\lambda ^{\beta _{n - 1} } |...|\lambda ^{\beta _2 } |\lambda ^{\beta _1 }\) be the (given) invariant factors of the square matricesA, resp.B of ordern over the ring of germs of holomorphic functions in 0 such that detA(λ)B(λ)≠0, λ≠0. A description of all possible invariant factors\(\lambda ^{\gamma _n } |\lambda ^{\gamma _{n - 1} } |...|\lambda ^{\gamma _2 } |\lambda ^{\gamma _1 }\) of the productC=AB is given in the following cases: (i)β 1 (or α1)≤2; (ii)β 3 = 0 (α3= 0); (iii) α1−α2, β1β m ≤1,α 2+1β m+1−0. These results, which hold for arbitraryn, are complemented with a few results leading to the description of all possible exponents γ1234 for arbitrary α1234 β1234 in the case where the ordern≤4.

1991 Mathematics Subject Classification

15A54