A rank-two spherical transform for eigenvector and singular-vector overlaps

Pierre Bousseyroux

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arXiv · 2609.22334

A rank-two spherical transform for eigenvector and singular-vector overlaps

Pierre Bousseyroux

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Abstract

We define a rank-two extension $\mathcal{H}^{(2)}$ of the spherical $\mathcal{H}$-transform, together with its associated transform $\mathcal{R}^{(2)}$. For sums, both transforms are additive. The transform $\mathcal{R}^{(2)}$ yields a $4\times4$ formula for the average product of two hermitized resolvents, and hence for off-diagonal eigenvector overlaps, recovering the Ginibre, elliptic, and bi-invariant formulas. We also treat correlated pairs of matrices, and obtain overlaps between their eigenvectors or singular vectors.

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Pierre Bousseyroux. 2026-09-16. A rank-two spherical transform for eigenvector and singular-vector overlaps. https://arxiv.org/abs/2609.22334

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