  Invariance transformations of polyadic decompositions of matrix
multiplication tensors define an equivalence relation on the set of such
decompositions. In this paper, we present an algorithm to efficiently decide
whether two polyadic decompositions of a given matrix multiplication tensor are
equivalent. With this algorithm, we analyze the equivalence classes of
decompositions of several matrix multiplication tensors. This analysis is
relevant for the study of fast matrix multiplication as it relates to the
question of how many essentially different fast matrix multiplication
algorithms there exist. This question has been first studied by de~Groote, who
showed that for the multiplication of $2\times2$ matrices with $7$ active
multiplications, all algorithms are essentially equivalent to Strassen's
algorithm. In contrast, the results of our analysis show that for the
multiplication of larger matrices, (e.g., $2\times3$ by $3\times2$ or
$3\times3$ by $3\times3$ matrices), two decompositions are very likely to be
essentially different. We further provide a necessary criterion for a polyadic
decomposition to be equivalent to a polyadic decomposition with integer
entries. Decompositions with specific integer entries, e.g., powers of two,
provide fast matrix multiplication algorithms with better efficiency and
stability properties. This condition can be tested algorithmically and we
present the conclusions obtained for the decompositions of small/medium matrix
multiplication tensors.
