Matrices over (algebraically closed) field can be block-diagonalized by conjugation,
each block being a Jordan cell. Matrices over a principal ideal domain (e.g. power
series in one variable) admit the diagonal reduction by left-right equivalence, A->
UAV. (The well known Smith normal form.) Over more general rings, e.g. power series in
several variables, most matrices do not admit such diagonal reduction. The simplest
obstruction is the irreducibility of the determinant.
Suppose that the determinant factorizes, what are the necessary/sufficient conditions
to ensure the corresponding block-diagonal reduction? Equivalently, assuming the
support of a module is reducible, when is the module the direct sum of the
corresponding components?
We give the complete answer for square matrices, assuming co-primeness of the
factorization.
For rectangular matrices the criterion is more technical.
As an immediate application we give criteria of simultaneous (block-)diagonal
reduction for tuples of matrices over a field, the splitting of determinantal
representations, the decomposability of sheaves on chains of curves, and so on.
Joint work with V. Vinnikov.
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