Problems Related to Matrix Multiplication

Abstract

Fast matrix multiplication algorithms play a fundamental role in the design of algorithmic solutions to several problems in computational linear algebra. In this chapter we will show that the complexities of problems such as matrix inversion, computation of the determinant, LUP-decomposition, computing the characteristic polynomial, or orthogonal basis transform, are dominated asymptotically by the complexity of matrix multiplication. Though we shall restrict ourselves to the muliplicative complexity only, the algorithms we exhibit show that these upper bounds also hold for the total complexity. On the other hand, matrix multiplication can be reduced to special instances of these problems, which shows that — from a computational point of view — all these problems are asymptotically equivalent. To put this sort of reasoning into a formal framework, we shall begin by introducing the notion of an exponent for a certain type of problems. We then show, by exhibiting specific algorithms, that all the above problems (and some others) have the same exponent as matrix multiplication. In the last section of this chapter we show how fast matrix multiplication algorithms can be used to compute the transitive closure of a graph.