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Principal Minors and Absolute Values

  • Francisco Villacis
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Abstract

We prove that the principal minors of an orthogonal projection matrix determine the coordinatewise absolute value of its image. More precisely, if W1,W2⊂ ℂn are subspaces whose orthogonal projection matrices have equal corresponding principal minors of all orders, then |W1|=|W2|, where |W| is the image of W under the coordinatewise absolute value map. The proof is divided into different cases related to the connectivity of the associated matroids. We provide a stratification of the Grassmannian in terms of the connectivity levels and give a structure theorem for the elements of each stratum.

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Cite

BibTeXRIS
BibTeX
@misc{villacis2026principal,
  title = {{Principal Minors and Absolute Values}},
  author = {Francisco Villacis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25499},
  eprint = {2609.25499},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25499},
}
RIS
TY  - UNPB
TI  - Principal Minors and Absolute Values
AU  - Francisco Villacis
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25499
UR  - https://arxiv.org/abs/2609.25499
AB  - We prove that the principal minors of an orthogonal projection matrix determine the coordinatewise absolute value of its image. More precisely, if W_1,W_2⊂ ℂ^n are subspaces whose orthogonal projection matrices have equal corresponding principal minors of all orders, then |W_1|=|W_2|, where |W| is the image of W under the coordinatewise absolute value map. The proof is divided into different cases related to the connectivity of the associated matroids. We provide a stratification of the Grassmannian in terms of the connectivity levels and give a structure theorem for the elements of each stratum.
ER  -