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arXiv · 2609.29177

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 $n\times n$ 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.

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Yuval Wigderson. 2026-09-24. Duality for matrix space questions. https://arxiv.org/abs/2609.29177

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