Feasible Proofs of Matrix Properties with Csanky's Algorithm
<|reference_start|>Feasible Proofs of Matrix Properties with Csanky's Algorithm: 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).<|reference_end|>
@article{soltys2005feasible,
    title={Feasible Proofs of Matrix Properties with Csanky's Algorithm},
    author={Michael Soltys},
    journal={arXiv preprint arXiv:cs/0505087},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0505087},
    primaryClass={cs.LO}
}