Permanental inequalities for correlation matrices

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}...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 1988-04, Vol.9 (2), p.194-201
Main Authors: GRONE, R, PIERCE, S
Format: Article
Language:English
Subjects:
Eigenvalues
Exact sciences and technology
Inequality
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Theorems
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Summary: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.
ISSN:0895-4798
1095-7162
DOI:10.1137/0609016