Permanental Inequalities for Correlation Matrices
Robert D. Grone, Stephen J. Pierce · SIAM Journal on Matrix Analysis and Applications · 1988
Let A be a positive semidefinite Hermitian matrix of order n with $|a_{11} | = \cdots = |a_{nn} | = 1$. We prove that $\operatorname{per} ( A )\geqq (1/n ) \| A \|^2 $, where $\| A \|$ is the Frobenius norm of A. This follows from a stronger result when $n = 4$, namely per $( A )\geqq \frac{1}{3} ( \| A \|^2 - 1 )$. Various corollaries are obtained.