Matrix multiplication: an open problem?
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In fact, a great many scientific fields require the multiplication of very large matrices (see Les Matrices, Bibliothèque Tangente 44, 2012). The basic algorithm for multiplying a matrix A with m rows and n columns by a matrix B with n rows and p columns (see opposite) requires m × n × p multiplications and n × p additions. On a computer, however, multiplications are considerably more "expensive" than additions. Mathematicians have therefore sought to reduce the number of multiplications, even at the cost of increasing the number of additions. This is what makes matrix multiplication an open problem. In 1967, for example, the German mathematician Volker Strassen (born in 1936) developed an algorithm that multiplies two square matrices faster than the basic method (see Les Algorithmes, Bibliothèque Tangente 37, 2013, and Tangente 189, 2019). This purely computational problem is precisely the one tackled by the team at DeepMind, Google's artificial intelligence (AI) subsidiary.
An example of matrix multiplication
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The product of two matrices follows a rule that may seem somewhat bewildering at first (it certainly does not involve multiplying the entries of the two matrices term by term!). Here, a numerical example is worth more than a lengthy formal treatment. Take the following matrix A, with two rows and three columns, and the matrix B below, with three rows and two columns.
