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PublicEdition 2

Sharp limitations of two permanental cofactor comparisons

Scott Schubr

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Abstract

We study two auxiliary cofactor comparisons motivated by permanent Hadamard inequalities. For a complex positive definite correlation matrix B, let R be its transposed permanental cofactor matrix and L that of its entrywise squared modulus. We compare z^TLz and z^T|R|^( ∘ 2)z, where z_i=|x_i|^2, with (per B)(x^*B^(-1)x)(x^*Rx). Both comparisons have sharp universal constant one in orders one and two. At every fixed order at least three, neither admits any finite universal constant, even among dense irreducible positive definite correlation matrices with nonreal triangle phases. An explicit roots-of-unity family gives quantitative divergence.

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