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

  • Pierre Bousseyroux
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Abstract

We define a rank-two extension ℋ(2) of the spherical ℋ-transform, together with its associated transform ℛ(2). For sums, both transforms are additive. The transform ℛ(2) yields a 4×4 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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Cite

BibTeXRIS
BibTeX
@misc{bousseyroux2026ranktwo,
  title = {{A rank-two spherical transform for eigenvector and singular-vector overlaps}},
  author = {Pierre Bousseyroux},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22334},
  eprint = {2609.22334},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22334},
}
RIS
TY  - UNPB
TI  - A rank-two spherical transform for eigenvector and singular-vector overlaps
AU  - Pierre Bousseyroux
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.22334
UR  - https://arxiv.org/abs/2609.22334
AB  - We define a rank-two extension ℋ^(2) of the spherical ℋ-transform, together with its associated transform ℛ^(2). For sums, both transforms are additive. The transform ℛ^(2) yields a 4×4 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.
ER  -