Equivalent polyadic decompositions of matrix multiplication tensors (Q2074879)

The rank of a tensor \(T\) is the minimal number \(r\) such that \(T=v_1+\dots+v_r\), where \(v_1,\dots,v_r\) are decomposable tensors. We can associate a tensor \(T_{m,n,p}\) to the bilinear form corresponding to the product of two matrices \(m\times n\) and \(n\times p.\) A lower bound on the rank \(r_{m,n,p},\) and the corresponding decomposition of \(T_{m,n,p},\) of such tensor enables one to construct algorithms for the matrix multiplication using less scalar multiplications and therefore the computations are more feasible. The importance of studying this tensor \(T_{m,n,p}\) and its rank is then well motivated. We say that two decompositions \(T_{n,m,p}=v_1+\dots+v_r\) and \(T_{n,m,p}=w_1+\dots+w_r\) are equivalent when, roughly speaking, the \(v_j\) can be moved simultaneously to the \(w_j\) by a change of coordinates. In the present paper, the authors provide an efficient algorithm to decide whether two decompositions of \(T_{n,m,p}\) are equivalent. It was already known that two decompositions of \(T_{2,2,2}\) are always equivalent. As an application of the algorithm the authors show by means of numerical experiments that two decompositions for \(T_{2,3,2}, T_{3,2,3}\) or \(T_{3,3,3}\) are most likely not equivalent.