Skip to main content

Some computational problems in linear algebra as hard as matrix multiplication

Abstract

We define the complexity of a computational problem given by a relation using the model of computation trees together with the Ostrowski complexity measure. Natural examples from linear algebra are:

  • KER n : Compute a basis of the kernel for a givenn×n-matrix,

  • OGB n : Find an invertible matrix that transforms a given symmetricn×n-matrix (quadratic form) into diagonal form,

  • SPR n : Find a sparse representation of a givenn×n-matrix.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    A. V. Aho, J. E. Hopcroft, andJ. D. Ullman,The design and analysis of computer algorithms, Reading MA: Addison-Wesley, 1974.

    Google Scholar 

  2. [2]

    A. Alder andV. Strassen,On the algorithmic complexity of associative algebras, Theor. Computer Science15 (1981), 201–211.

    Google Scholar 

  3. [3]

    W. Baur andV. Strassen,The complexity of partial derivatives. Theor. Computer Science22 (1982), 317–330.

    Google Scholar 

  4. [4]

    L. Blum, M. Shub, andS. Smale,On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc.21 (1989), 1–46.

    Google Scholar 

  5. [5]

    J. Bunch andJ. Hopcroft,Triangular factorization, and inversion by fast matrix multiplication, Math. Comp.28 (1974), 231–236.

    Google Scholar 

  6. [6]

    D. Coppersmith andS. Winograd,Matrix multiplication via arithmetic progressions, J. Symb. Comp.9 (1990), 251–280.

    Google Scholar 

  7. [7]

    R. Hartshorne,Algebraic Geometry, Graduate Texts in Mathematics Vol. 52, Springer Verlag, 1977.

  8. [8]

    K. Kalorkoti,The trace invariant and matrix inversion. Theor. Computer Science59 (1988), 277–286.

    Google Scholar 

  9. [9]

    W. Keller-Gehrig,Fast algorithms for the characteristic polynomial, Theor. Computer Science36 (1985), 309–317.

    Google Scholar 

  10. [10]

    H. Kraft,Geometric methods in representation theory, in: Representations of Algebras, Workshop Proc., Puebla, Mexico 1980, LNM944, Berlin-Heidelberg-New York 1982.

  11. [11]

    J. C. Lafon and S. Winograd,A lower bound for the multiplicative complexity of the product of two matrices, (unpublished) manuscript, 1978.

  12. [12]

    T. Lickteig,On semialgebraic decision complexity, Tech. Rep. TR-90-052 Int. Comp. Science Inst., Berkeley, and Univ. Tübingen, Habilitationsschrift, to appear.

  13. [13]

    A. Schönhage,Unitäre Transformationen grosser Matrizen, Num. Math.20 (1973), 409–417.

    Google Scholar 

  14. [14]

    V. Strassen,Gaussian elimination is not optimal, Numer. Mathematik13 (1969), 354–356.

    Google Scholar 

  15. [15]

    V. Strassen,Berechnung und Programm I, Acta Informatica1 (1973), 320–335.

    Google Scholar 

  16. [16]

    V. Strassen,Berechnung und Programm II, Acta Informatica2 (1973), 64–79.

    Google Scholar 

  17. [17]

    V. Strassen,Vermeidung von Divisionen, Crelles Journal für die reine und angewandte Mathematik264 (1973), 184–202.

    Google Scholar 

  18. [18]

    V. Strassen,The complexity of continued fraction, SIAM J. Comp.12/1 (1983), 1–27.

    Google Scholar 

  19. [19]

    V. Strassen,Relative bilinear complexity and matrix multiplication, J. für die reine und angewandte Mathematik375/376 (1987), 406–443.

    Google Scholar 

  20. [20]

    I. Wegener,The complexity of Boolean functions, Wiley-Teubner, 1987.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bürgisser, P., Karpinski, M. & Lickteig, T. Some computational problems in linear algebra as hard as matrix multiplication. Comput Complexity 1, 131–155 (1991). https://doi.org/10.1007/BF01272518

Download citation

Key words

  • Problems
  • computation trees
  • straight line programs
  • Ostrowski complexity
  • derivations
  • matrix multiplication

Subject classifications

  • 68C20
  • 68C25