Feasible Proofs of Matrix Properties with Csanky’s Algorithm
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We show that Csanky's fast parallel algorithm for computing the
characteristic polynomial of a matrix can be formalized in the logical theory
LAP, and can be proved correct in LAP from the principle of linear
independence. LAP is a natural theory for reasoning about linear algebra
introduced by Cook and Soltys. Further, we show that several principles of
matrix algebra, such as linear independence or the Cayley-Hamilton Theorem, can
be shown equivalent in the logical theory QLA. Applying the separation between
complexity classes AC^0[2] contained in DET(GF(2)), we show that these
principles are in fact not provable in QLA. In a nutshell, we show that linear
independence is ``all there is'' to elementary linear algebra (from a proof
complexity point of view), and furthermore, linear independence cannot be
proved trivially (again, from a proof complexity point of view).