Isotropic Decompositions via Inverse Eigenvectors

Abstract

We develop a residue-theoretic framework for studying inverse eigenvectors of a square matrix, defined by the nonlinear equation $Mα=α^{-1}$. Our main result is an inverse analogue of the spectral theorem: under natural transversality and properness assumptions, the identity operator admits an explicit decomposition into rank-one tensors associated with the inverse eigenvectors. The proof is based on residues of rational differential forms in several complex variables. For real correlation matrices whose off-diagonal entries have modulus strictly less than one, we remove the properness assumption and show that the coefficients in the decomposition are positive and sum to~$1$. Consequently, the inverse eigenvectors support an explicit centered discrete isotropic probability measure and form a weighted tight frame. We extend the construction to arbitrary real Gram matrices, diagonal inverse eigenvectors, and weighted inverse-eigenvector equations. Simple trace and arithmetic--geometric mean arguments yield inverse eigenvectors with controlled Euclidean norm and coordinate product. These estimates lead to short proofs of the strong real polarization inequality and the $n$th real linear polarization inequality, together with weighted and matrix-valued generalizations and further geometric and analytic applications.

Disclosure

“ance during the author’s doctoral studies at UCL, and Péter Frenkel, Máté Matolcsi, Oscar Ortega-Moreno, and Szilárd Révész for valuable conversations on polarization problems. During the preparation of this manuscript, the author used ChatGPT (GPT-5.6 Think- ing) as a research assistant.”

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2026-08-23T19:11:19.508415Z

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