For random 7x7 matrices with entries uniform in (-1,1), the ratio of the two lowest characteristic-polynomial coefficients, equal to 1/tr(A^{-1}) for monotone A, is empirically below 0.1755 for all 18,000 sampled monotone matrices.
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Explaining Deep Network Classification of Matrices: A Case Study on Monotonicity
For random 7x7 matrices with entries uniform in (-1,1), the ratio of the two lowest characteristic-polynomial coefficients, equal to 1/tr(A^{-1}) for monotone A, is empirically below 0.1755 for all 18,000 sampled monotone matrices.