Abstract
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 in [8]. 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 \(\textbf{AC}^0[2]\subsetneq\textbf{DET}(\text{GF}(2))\), 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).
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Soltys, M. (2005). Feasible Proofs of Matrix Properties with Csanky’s Algorithm. In: Ong, L. (eds) Computer Science Logic. CSL 2005. Lecture Notes in Computer Science, vol 3634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538363_34
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DOI: https://doi.org/10.1007/11538363_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28231-0
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