On the complexity of bilinear forms with commutativity
J Ja'Ja' - Proceedings of the eleventh annual ACM symposium …, 1979 - dl.acm.org
J Ja'Ja'
Proceedings of the eleventh annual ACM symposium on Theory of computing, 1979•dl.acm.orgWe consider the problem of computing a set of bilinear forms in the case when the
indeterminates commute. We develop lower bound techniques which seem to be more
powerful than those already known in the literature for the commutative case. An unexpected
result is the fact that Duality theory does not hold in the commutative case; we prove that the
multiplication of 2× n by n× 2 matrices requires at least [27n/8] multiplications while it is
possible to multiply 2× 2 by 2× n matrices using only 3n+ 2 multiplications. We also settle the …
indeterminates commute. We develop lower bound techniques which seem to be more
powerful than those already known in the literature for the commutative case. An unexpected
result is the fact that Duality theory does not hold in the commutative case; we prove that the
multiplication of 2× n by n× 2 matrices requires at least [27n/8] multiplications while it is
possible to multiply 2× 2 by 2× n matrices using only 3n+ 2 multiplications. We also settle the …
We consider the problem of computing a set of bilinear forms in the case when the indeterminates commute. We develop lower bound techniques which seem to be more powerful than those already known in the literature for the commutative case. An unexpected result is the fact that Duality theory does not hold in the commutative case; we prove that the multiplication of 2 × n by n × 2 matrices requires at least [27n/8] multiplications while it is possible to multiply 2 × 2 by 2 × n matrices using only 3n + 2 multiplications. We also settle the question of whether commutativity can reduce the number of multiplications by 1/2 by showing that this can never happen.
On the other hand, we show that, over algebraically closed fields, the complexity of computing a pair of bilinear forms is the same whether or not commutativity is allowed. We feel that, in general, commutativity will have little effect whenever the constant set is an algebraically closed field.

Showing the best result for this search. See all results