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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Extending the Hölder type inequality of Blakley and Roy to non-symmetric non-square matrices


Author: Thomas H. Pate
Journal: Trans. Amer. Math. Soc. 364 (2012), 4267-4281
MSC (2010): Primary 15A63, 15A42, 15A60, 15A15, 05C50
Published electronically: March 20, 2012
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Abstract: Suppose $ m$, $ n$, and $ k$ are positive integers, and let $ \langle \cdot ,\cdot \rangle $ denote standard inner product on the spaces $ \mathbb{R}^p$, $ p\!>\!0$. We show that if $ D$ is an $ m\!\times \!n$ non-negative real matrix, and $ u$ and $ v$ are non-negative unit vectors in $ \mathbb{R}^n$ and $ \mathbb{R}^m$, respectively, then

$\displaystyle \langle (D\!D^t)^k D u ,v \rangle \,\ge \, \langle Du,v \rangle ^{2k+1},$ (1)

with equality if and only if $ \langle (DD^t)^k D u ,v \rangle = 0$, or there exists $ \alpha > 0$ such that $ Du = \alpha v$ and $ D^t v = \alpha u$. This inequality extends to non-symmetric non-square matrices a 1965 result of Blakley and Roy which asserts that if $ D$ is a non-negative $ n\!\times \!n$ symmetric matrix, and $ u\!\in \!\mathbb{R}^n$ is a non-negative unit vector, then

$\displaystyle \langle D^k u, u \rangle \, \ge \,\langle Du,u \rangle ^{k},$ (2)

with equality, when $ k\!\ge \!2$, if and only if $ \langle D^k u, u \big \rangle = 0$, or there exists $ \alpha \!>\!0$ such that $ Du = \alpha u$. The generality of the inequality (1) derives not only from the fact that $ D$ is not assumed to be symmetric or square, but from the fact that we admit two unit vectors $ u$ and $ v$ instead of the single unit vector $ u$ appearing in inequality (2) of Blakley and Roy. We apply our result to verify the conjecture of A. Sidorenko (1993) in the non-symmetric case provided that the underlying graph is a path.

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Additional Information

Thomas H. Pate
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05501-2
PII: S 0002-9947(2012)05501-2
Keywords: Positive matrices, inequalities, Hölder’s inequality, bipartite graphs
Received by editor(s): August 20, 2010
Received by editor(s) in revised form: November 5, 2010
Published electronically: March 20, 2012
Article copyright: © Copyright 2012 American Mathematical Society