Invertible sums of matrices

Justin Chen

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arXiv · 1603.06696

Invertible sums of matrices

Justin Chen

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Abstract

We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli. The proof yields a polynomial identity, expressing the determinant of a large sum of matrices in terms of determinants of smaller sums. Interpreting these results over an arbitrary commutative ring gives a stabilization result for a filtered family of ideals of determinants. Generalizing in another direction gives a characterization of local rings. An analogous result for semilocal rings is also given -- interestingly, the semilocal case reduces to the case of matrices.

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Justin Chen. 2016-04-19. Invertible sums of matrices. https://arxiv.org/abs/1603.06696

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