A large class of multiplication problems in arithmetic complexity can be viewed as the simultaneous evaluation of a set of bilinear forms. This class includes the multiplication of matrices, polynomials, quaternions, Cayley and complex numbers. Considering bilinear algorithms, the optimal number of nonscalar multiplications can be described as the rank of a three-tensor or as the smallest member of rank one matrices necessary to include a given set of matrices in their span.
In this paper, we attack a rather large subclass of three-tensors, namely that of $(p,q,2)$-tensors, for arbitrary p and q, and solve it completely in the case where the field of constants contains the roots of a polynomial associated with the given tensor. In all other cases, we prove that, in general, our bounds cannot be improved. The complexity of a general pair of bilinear forms is determined explicitly in terms of parameters related to Kronecker’s theory of pencils and to the theory of invariant polynomials. This reveals unexpected results and shows explicitly the dependence on the algebraic structure of the constants; we display, for example, a pair of $3 \times 3$ bilinear forms whose complexity is 3 over the field $Z_7 $ and which, however, requires exactly 4 nonscalar multiplications over the fields $Z_5 $ or $Z_{11} $. Corresponding optimal algorithms are described and several applications are considered.


  1. algebraic complexity
  2. bilinear forms
  3. tensor rank

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Alfred V. Aho, John E. Hopcroft, Jeffrey D. Ullman, The design and analysis of computer algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975x+470
M. D. Atkinson, N. M. Stephens, On the maximal multiplicative complexity of a family of bilinear forms, Linear Algebra Appl., 27 (1979), 1–8
M. D. Atkinson, private communication
Roger W. Brockett, David Dobkin, On the optimal evaluation of a set of bilinear forms, Linear Algebra and Appl., 19 (1978), 207–235
Roger Brockett, David Dobkin, On the number of multiplications required for matrix multiplication, SIAM J. Comput., 5 (1976), 624–628
D. Dobkin, Masters Thesis, On the arithmetic complexity of a class of arithmetic computations, Thesis, Harvard University, Cambridge, MA, 1973, September
Charles M. Fiduccia, R. Miller, J. Thatcher, On obtaining upper bounds on the complexity of matrix multiplicationComplexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, 31–40, 187–212
C. M. Fiduccia, Y. Zalcstein, Algebras having linear multiplicative complexities, Technical Report no. 46, Department of Computer Science, State University of New York at Stony Brook, 1975, August
F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959Vol. 1, x+374 pp. Vol. 2, ix+276
N. Gastinel, Sur le calcul des produits de matrices, Numer. Math., 17 (1971), 222–229
N. Gastinel, Le Problèms De L'Extension Minimale Diagonals D'un Operateur Linéaire, no. 235, 1975, manuscript communicated by one of the referees
D. Grigor'ev, N. K. Nikol'sku, On the algebraic complexity of computing a pair of bilinear formsInvestigations on Linear Operators, Izdat. “Nauka” Leningrad Otdel., Leningrad, 1974, 159–163
J. E. Hopcroft, L. R. Kerr, On minimizing the number of multiplications necessary for matrix multiplication, SIAM J. Appl. Math., 20 (1971), 30–36
J. Hopcroft, J. Musinski, Duality applied to the complexity of matrix multiplication and other bilinear forms, SIAM J. Comput., 2 (1973), 159–173
T. D. Howell, Ph.D. Thesis, Tensor rank and the complexity of bilinear forms, Cornell University, 1976, Sept.
T. D. Howell, J. C. Lafon, The complexity of the quaternionproduct, TR, 75-245, Department of Computer Science, Cornell University, 1975, June
J. Ja' Ja', Ph.D. Thesis, On the algebraic complexity of classes of bilinear forms, Harvard University, 1977, Sept.
J. Ja' Ja', Computation of bilinear forms over finite fields, Technical Report, CS-78-03, Department of Computer Science, Pennsylvania State University, 1978, Jan.
Julian D. Laderman, A noncommutative algorithm for multiplying $3\times 3$ matrices using $23$ muliplications, Bull. Amer. Math. Soc., 82 (1976), 126–128, January
Jean-Claude Lafon, Optimum computation of p bilinear forms, Linear Algebra and Appl., 10 (1975), 225–240
Ju. V. Matijasevič, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR, 191 (1970), 279–282
I. Munro, Problems related to matrix multiplication, Proceedings Courant Institute Symposium on Computational Complexity, New York, 1971, Oct.
Morris Newman, Integral matrices, Academic Press, New York, 1972xvii+224
R. Probert, On the complexity of symmetric computations, Technical Report, CS-73-02, University of Waterloo Computer Science, Waterloo, Ontario, 1973, Jan.
Volker Strassen, Gaussian elimination is not optimal, Numer. Math., 13 (1969), 354–356
Volker Strassen, R. Miller, J. Thatcher, Evaluation of rational functionsComplexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, 1–10, 187–212
Volker Strassen, Vermeidung von Divisionen, J. Reine Angew. Math., 264 (1973), 184–202
S. Winograd, On multiplication of $2\times 2$ matrices, Linear Algebra and Appl., 4 (1971), 381–388

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Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 443 - 462
ISSN (online): 1095-7111


Submitted: 8 November 1977
Published online: 13 July 2006


  1. algebraic complexity
  2. bilinear forms
  3. tensor rank



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