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Abstract

For a complex correlation matrix B of order n, let C(B)_{ij}=per B(i| j) be its raw permanental cofactor matrix. We bound the optimal constant γ_n in C(B)≼γ_n(per B)I. An elementary isolated-row induction and an explicit positive-definite equicorrelation family give, for every n≥2, ((n+1)^2)/(4n)<γ_n≤(n^2+2n+3)/(4n). Consequently γ_n=n/4+1/2+Θ(1/n): both the leading coefficient and the constant term are sharp. The upper estimate extends, with the necessary diagonal weights, to all Hermitian positive semidefinite matrices, including zero diagonals. We also obtain an explicit nonnegative reserve for the standard natural-representation block in order six, and give an exact order-six obstruction to the raw cofactor bound with constant two. The proof is analytic and does not require finite polynomial certificates.