The proof complexity of linear algebra

The Proof Complexity of Linear Algebra

We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley-Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB = I → BA = I.

Proof Complexity Meets Algebra

We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional and semi-algebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes...

A linear algebra proof of the fundamental theorem of algebra

We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional vector space admits eigenvectors. The argument avoids the use of the fundamental theorem of algebra, which can then be deduced from it. Our presentation avoids any appeal to the theory of determinants. 2010 Mathematics Subject Classification. 15-01, 15A18.

On the Descriptive Complexity of Linear Algebra

The Proof Complexity of Linear AlgebraMi