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From the classic work of Gohberg and Krein (1958), it is well known that theset of partial indices of a non-singular matrix function may change dependingon the properties of the original matrix. More precisely, it was shown that ifthe difference between the larger and the smaller partial indices is largerthan unity then, in any neighborhood of the original matrix function, thereexists another matrix function possessing a different set of partial indices.As a result, the factorization of matrix functions, being an extremelydifficult process itself even in the case of the canonical factorization,remains unresolvable or even questionable in the case of a non-stable set ofpartial indices. Such a situation, in turn, has became an unavoidable obstacleto the application of the factorization technique. This paper sets out toanswer a less ambitious question than that of effective factorizing matrixfunctions with non-stable sets of partial indices, and instead focuses ondetermining the conditions which, when imposed on the original matrix function,allow to construct another matrix function that exhibits the same partialindices and is close to the original matrix function.

Regular approximate factorization of a class of matrix-function with an unstable set of partial indices