Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2012-05501-2
Published electronically: March 20, 2012
MathSciNet review: 2912454
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 \begin{equation} \langle (D\!D^t)^k D u ,v \rangle \ge \langle Du,v \rangle ^{2k+1}, \end{equation} 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 \begin{equation} \langle D^k u, u \rangle \ge \langle Du,u \rangle ^{k}, \end{equation} 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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 15A63, 15A42, 15A60, 15A15, 05C50

Retrieve articles in all journals with MSC (2010): 15A63, 15A42, 15A60, 15A15, 05C50


Additional Information

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

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