Addition Requirements for Matrix and Transposed Matrix Products

Let M be an s ×t matrix and let M T be the transpose of M . Let x and y be t − and s −dimensional indeterminate column vectors, respectively. We show that any linear algorithm A that computes M x has associated with it a natural dual linear algorithm denoted A T that computes M T y . Furthermore, if M has no zero rows or columns then the number of additions used by A T exceeds the number of additions used by A by exactly s −t . In addition, a strong correspondence is established between linear algorithms that compute the product M x and bilinear algorithms that compute the bilinear form y T M x .