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Duality for matrix space questions

  • Yuval Wigderson
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Abstract

We present a new proof of a classical theorem of Dieudonné: if a linear space of n× n matrices consists entirely of singular matrices, then its dimension is at most n2−n. Our proof is based on a surprising ``duality'' argument: we prove this universal upper bound by exhibiting a single matrix space that serves as a lower bound for a related problem. Interestingly, this approach only works for certain fields, but we use model-theoretic arguments to obtain the same result for all fields. We hope that this approach can be generalized to provide new duality-based proofs of other classical theorems on matrix spaces, and give some preliminary results in this direction.

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Cite

BibTeXRIS
BibTeX
@misc{wigderson2026duality,
  title = {{Duality for matrix space questions}},
  author = {Yuval Wigderson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29177},
  eprint = {2609.29177},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29177},
}
RIS
TY  - UNPB
TI  - Duality for matrix space questions
AU  - Yuval Wigderson
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29177
UR  - https://arxiv.org/abs/2609.29177
AB  - We present a new proof of a classical theorem of Dieudonné: if a linear space of n× n matrices consists entirely of singular matrices, then its dimension is at most n^2−n. Our proof is based on a surprising ``duality'' argument: we prove this universal upper bound by exhibiting a single matrix space that serves as a lower bound for a related problem. Interestingly, this approach only works for certain fields, but we use model-theoretic arguments to obtain the same result for all fields. We hope that this approach can be generalized to provide new duality-based proofs of other classical theorems on matrix spaces, and give some preliminary results in this direction.
ER  -