LA, permutations, and the Hajós Calculus

LA is a simple and natural logical system for reasoning about matrices. We show that LA, over finite fields, proves a host of matrix identities (so-called “hard matrix identities”) from the matrix form of the pigeonhole principle. LAP is LA with matrix powering; we show that LAP extended with quantification over permutations is strong enough to prove fundamental theorems of linear algebra (such as the Cayley-Hamilton Theorem). Furthermore, we show that LA with quantification over permutations expresses NP graph-theoretic properties, and proves the soundness of the Hajós Calculus. Several open problems are stated.