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Proving properties of matrices over \({\mathbb{Z}_{2}}\)

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Abstract

We prove assorted properties of matrices over \({\mathbb{Z}_{2}}\), and outline the complexity of the concepts required to prove these properties. The goal of this line of research is to establish the proof complexity of matrix algebra. It also presents a different approach to linear algebra: one that is formal, consisting in algebraic manipulations according to the axioms of a ring, rather than the traditional semantic approach via linear transformations.

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References

  1. Albert A.A.: Symmetric and alternating matrices in an arbitrary field, i. Trans. Am. Math. Soc. 43(3), 386–436 (1938)

    Google Scholar 

  2. Brualdi R.A., Ryser H.J.: Combinatorial Matrix Theory. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  3. Cook, S., Fontes, L.: Formal theories for linear algebra. In: Presented at the Federated Logic Conference (2010)

  4. Cobb S.M.: On powers of matrices with elements in the field of integers modulo 2. Math. Gazette 42(342), 267–271 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  5. Filmus, Y.: Range of symmetric matrices over GF(2). Unpublished note, University of Toronto, January (2010)

  6. MacWilliams J.: Orthogonal matrices over finite fields. Am. Math. Mon. 76(2), 152–164 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Montgomery, P.L.: A block Lanczos algorithm for finding dependencies over GF(2). In: EUROCRYPT pp. 106–120 (1995)

  8. Ryser H.J.: Matrices of zeros and ones. Bull. Am. Math. Soc. 66(6), 442–464 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  9. Soltys M., Cook S.A.: The complexity of derivations of matrix identities. Ann. Pure Appl. Logic 130(1–3), 277–323 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Soltys M.: Extended frege and Gaussian elimination. Bull. Sect. Logic 31(4), 1–17 (2002)

    MathSciNet  Google Scholar 

  11. Thapen N., Soltys M.: Weak theories of linear algebra. Arch. Math. Logic 44(2), 195–208 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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  1. McMaster University, Hamilton, Canada

    Michael Soltys

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  1. Michael Soltys
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Correspondence to Michael Soltys.

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Soltys, M. Proving properties of matrices over \({\mathbb{Z}_{2}}\) . Arch. Math. Logic 51, 535–551 (2012). https://doi.org/10.1007/s00153-012-0280-0

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  • DOI: https://doi.org/10.1007/s00153-012-0280-0

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