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Improved lower bounds on the rigidity of Hadamard matrices. (English. Russian original) Zbl 0917.15013
Math. Notes 63, No. 4, 471-475 (1998); translation from Mat. Zametki 63, No. 4, 535-540 (1998).
It is proved that for any $$n\times n$$ generalized Hadamard matrix $$H$$ and any $$r\leq n/2$$ the inequality $$R_H^c(r)\geq \Omega(n^2/r)$$ holds. Moreover, if $$\theta$$ is an additional parameter satisfying $$\theta\geq n/r$$, then $$R_H^c(r,\theta)\geq \Omega(n^3/r\theta^2)$$. Here, $$R_H^c(r)$$ and $$R_H^c(r,\theta)$$ are the rigidity functions of $$H$$ while $$\Omega(g)=f$$ in $$f(x)\geq cg(x)$$ with some positive constant $$c$$ for all $$x$$ from the domain of the functions $$f$$ and $$g$$.

##### MSC:
 15A42 Inequalities involving eigenvalues and eigenvectors
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##### References:
 [1] L. G. Valiant,Some Conjectures Relating to Superlinear Complexity Bounds, Tech. Report No. 85, Univ. of Leeds (1976). [2] L. G. Valiant,Graph-Theoretic Arguments in Low-Level Complexity, Tech. Report No. 13-77, Univ. of Edinburgh, Dep. of Comp. Sci. (1977). [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). · Zbl 0341.94020 [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). · Zbl 0504.68024 [5] A. A. Razborov,On Rigid Matrices [in Russian], Manuscript (1989). [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). · Zbl 0937.68515 [7] J. Friedman, ”A note on matrix rigidity,”Combinatorica,13, No. 2, 235–239 (1993). · Zbl 0848.15005 · doi:10.1007/BF01303207 [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). [9] P. Kimmel and A. Settle,Reducing the Rank of Lower Triangular All-Ones Matrix, Tech. Report CS 92-21, Univ. of Chicago (1992). [10] P. Pudlák, ”Large communication in constant depth circuits,”Combinatorica,14, No. 2, 203–216 (1994). · Zbl 0819.68090 · doi:10.1007/BF01215351 [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). [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). · Zbl 1059.68577 [13] P. Pudlák, A. Razborov, and P. Savický,Observations on Rigidity of Hadamard Matrices [in Russian], Manuscript (1988). [14] N. Alon,On the Rigidity of Hadamard Matrices, Manuscript (1990). [15] A. J. Hoffman and H. W. Wielandt, ”The variation of the spectrum of a normal matrix,”Duke Math. J.,20, 37–39 (1953). · Zbl 0051.00903 · doi:10.1215/S0012-7094-53-02004-3 [16] G. H. Golub and C. F. van Loan,Matrix Computations, John Hopkins Univ. Press (1983). · Zbl 0559.65011 [17] B. S. Kashin, ”On some properties of matrices of bounded operators from the space 2 n into 2 n , ”Izv. Akad. Nauk Arm. SSR Mat.,15, No. 5, 379–394 (1980). · Zbl 0452.15023 [18] A. A. Lunin, ”Operator norms of submatrices,”Mat. Zametki [Math. Notes]45, No. 3, 248–252 (1989). · Zbl 0687.15021 [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). [20] M. Rudelson, ”Almost orthogonal submatrices of an orthogonal matrix,”Israel J. Math (to appear). · Zbl 0958.15019
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