Feasible Proofs of Matrix Properties with Csanky’s Algorithm

  • Michael Soltys
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)

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).

Keywords

Proof complexity Csanky’s algorithm matrix algebra 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Michael Soltys
    • 1
  1. 1.Computing and SoftwareMcMaster UniversityHamiltonCanada

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