Inverses of symmetric, diagonally dominant positive matrices and applications

Details Inverses of symmetric, diagonally dominant positive matrices and applications Inverses of symmetric, diagonally dominant positive matrices and applications

2012-03-29

arxiv.org/abs/1203.6812v1

Mathematics

Functional Analysis

17 pages

Scientific paper

We prove tight bounds for the $\infty$-norm of the inverse of a symmetric, diagonally dominant positive matrix $J$; in particular, we show that $\|J^{-1}\|_{\infty}$ is uniquely maximized among all such $J$. We also prove a new lower-bound form of Hadamard's inequality for the determinant of diagonally dominant positive matrices and an improved upper bound for diagonally balanced positive matrices. Applications of our results include numerical stability for linear systems, bounds on inverses of differentiable functions, and consistency of the maximum likelihood equations for random graph distributions.

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