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Classical matrix theory works over the complex numbers; its reliance on the usual absolute value makes it an Archimedean theory. In this paper, we consider non-Archimedean counterparts of the Gershgorin disk theorem and diagonally dominant matrices. We compare and contrast the Archimedean and non-Archimedean contexts. A remarkable dissimilarity is that diagonally dominant matrices enjoy more structure in the non-Archimedean setting.

Non-Archimedean Views on Gershgorin’s Theorem and Diagonal Dominance