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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Extending the Hölder type inequality of Blakley and Roy to non-symmetric non-square matrices
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by Thomas H. Pate PDF
Trans. Amer. Math. Soc. 364 (2012), 4267-4281 Request permission


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.
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Additional Information
  • Thomas H. Pate
  • Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
  • Received by editor(s): August 20, 2010
  • Received by editor(s) in revised form: November 5, 2010
  • Published electronically: March 20, 2012
  • © Copyright 2012 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 4267-4281
  • MSC (2010): Primary 15A63, 15A42, 15A60, 15A15, 05C50
  • DOI:
  • MathSciNet review: 2912454