Mathematical Notes

, Volume 63, Issue 4, pp 471–475 | Cite as

Improved lower bounds on the rigidity of Hadamard matrices

  • B. S. Kashin
  • A. A. Razborov


We writef=Ω(g) iff(x)≥cg(x) with some positive constantc for allx from the domain of functionsf andg. We show that at least Ω(n 2 /r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank belowr. This improves the previously known bound Ω(n2/r2). If we additionally know that the changes are bounded above in absolute value by some numberθ≥n/r, then the number of these entries is bounded below by Ω(n3/( 2 )), which improves upon the previously known bound Ω(n 2 /θ 2 ).

Key words

rigidity of matrices Hadamard matrices spectral methods Hoffman-Wielandt inequality 


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  1. 1.
    L. G. Valiant,Some Conjectures Relating to Superlinear Complexity Bounds, Tech. Report No. 85, Univ. of Leeds (1976).Google Scholar
  2. 2.
    L. G. Valiant,Graph-Theoretic Arguments in Low-Level Complexity, Tech. Report No. 13-77, Univ. of Edinburgh, Dep. of Comp. Sci. (1977).Google Scholar
  3. 3.
    D. Yu. Grigor'ev, “An application of separability and independence notions for obtaining lower bounds of circuit complexity,”Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (POMI),60, 38–48 (1976).zbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Yu. Grigor'ev, “Lower bounds in the algebraic computational complexity,”Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (POMI),118, 25–82 (1982).zbMATHMathSciNetGoogle Scholar
  5. 5.
    A. A. Razborov,On Rigid Matrices [in Russian], Manuscript (1989).Google Scholar
  6. 6.
    S. V. Lokam, “Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity,” in:Proc. of the 36th IEEE Symposium on Foundations of Computer Science, Los Alamitos (CA), 6–15 (1995).Google Scholar
  7. 7.
    J. Friedman, “A note on matrix rigidity,”Combinatorica,13, No. 2, 235–239 (1993).zbMATHMathSciNetGoogle Scholar
  8. 8.
    P. Pudlák and Z. Vavřín, “Computation of rigidity of ordern 2/r for one simple matrix,”Comment. Math. Univ. Carolin.,32, No. 2, 213–218 (1991).MathSciNetGoogle Scholar
  9. 9.
    P. Kimmel and A. Settle,Reducing the Rank of Lower Triangular All-Ones Matrix, Tech. Report CS 92-21, Univ. of Chicago (1992).Google Scholar
  10. 10.
    P. Pudlák, “Large communication in constant depth circuits,”Combinatorica,14, No. 2, 203–216 (1994).zbMATHMathSciNetGoogle Scholar
  11. 11.
    M. Krause and S. Waack, “Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in,” in:Proc. of the 32nd IEEE Symposium on Foundations of Computer Science, Los Alamitos (CA), 777–782 (1991).Google Scholar
  12. 12.
    N. Nisan and A. Wigderson, “On the complexity of bilinear forms,” in:Proc. of the 27th ACM Symposium on the Theory of Computing, New York, 723–732 (1995).Google Scholar
  13. 13.
    P. Pudlák, A. Razborov, and P. Savický,Observations on Rigidity of Hadamard Matrices [in Russian], Manuscript (1988).Google Scholar
  14. 14.
    N. Alon,On the Rigidity of Hadamard Matrices, Manuscript (1990).Google Scholar
  15. 15.
    A. J. Hoffman and H. W. Wielandt, “The variation of the spectrum of a normal matrix,”Duke Math. J.,20, 37–39 (1953).CrossRefMathSciNetGoogle Scholar
  16. 16.
    G. H. Golub and C. F. van Loan,Matrix Computations, John Hopkins Univ. Press (1983).Google Scholar
  17. 17.
    B. S. Kashin, “On some properties of matrices of bounded operators from the space ℓ2n into ℓ2n, ”Izv. Akad. Nauk Arm. SSR Mat.,15, No. 5, 379–394 (1980).zbMATHMathSciNetGoogle Scholar
  18. 18.
    A. A. Lunin, “Operator norms of submatrices,”Mat. Zametki [Math. Notes]45, No. 3, 248–252 (1989).zbMATHMathSciNetGoogle Scholar
  19. 19.
    B. Kashin and L. Tzaffiri,Some Remarks on the Restriction of Operators to Coordinate Subspaces, Tech. Report No. 12, The Edmund Landau Center for Research in Math. Anal., Hebrew Univ., Jerusalem (1993/94).Google Scholar
  20. 20.
    M. Rudelson, “Almost orthogonal submatrices of an orthogonal matrix,”Israel J. Math (to appear).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • B. S. Kashin
    • 1
  • A. A. Razborov
    • 1
  1. 1.V. A. Steklov Mathematics InstituteRussian Academy of SciencesUSSR

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