On the Exponent of Certain Matrix Operations

Abstract

A great deal of research was done in the period 1969–1987 on fast matrix operations, including [185, 212, 206, 213, 62]. Various proofs showed that many important matrix operations, such as QR-decomposition, LU-factorization, inversion, and finding determinants, are no more complex than matrix multiplication, in the big-Oh sense, see [13, Ch. 6] or [63, Ch. 28].

For this reason, many fast matrix multiplication algorithms were developed. Almost all were intended to work over a general ring. However, one in particular was intended for boolean matrices, and by extension $\mathbb{G}\mathbb{F}$(2)-matrices, which was named the Method of Four Russians, “after the cardinality and the nationality of its inventors.” While the Method of Four Russians was conceived as a boolean matrix multiplication tool, we show how to use it for $\mathbb{G}\mathbb{F}$(2) matrices and for inversion, in Section 9.3 on Page 137 and Section 9.4 on Page 141.