By distilling and combining earlier arguments, we give a self-contained and entirely elementary proof of the strong and real linear polarization inequalities established by Mart\'inez and Ortega-Moreno. Specifically, for any unit vectors $u_1,\ldots,u_n\in\mathbb{R}^d$, there exists a unit vector $v\in\mathbb{R}^d$ suc...
We give a self-contained and entirely elementary proof of the strong polarization inequality: for any unit vectors $u_1,\ldots,u_n\in\mathbb{R}^d$, there exists a unit vector $v\in\mathbb{R}^d$ such that \[ \sum_{i=1}^n \frac{1}{\langle u_i,v\rangle^2}\le n^2. \] As an immediate consequence, we recover the real linear...
We develop a residue-theoretic framework for studying inverse eigenvectors of a square matrix, defined by the nonlinear equation $M\alpha=\alpha^{-1}$. Our main result is an inverse analogue of the spectral theorem: under natural transversality and properness assumptions, the identity operator admits an explicit decomp...
Gergely Ambrus· 0 citations
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