  In this paper, the canonical polyadic (CP) decomposition of tensors that
corresponds to matrix multiplications is studied. Finding the rank of these
tensors and computing the decompositions is a fundamental problem of algebraic
complexity theory. In this paper, we characterize existing decompositions
(found by any algorithm) by certain vectors called signature, and transform
them in another decomposition which can be more suitable in practical
algorithms. In particular, we present a novel decomposition of the tensor
multiplication of matrices of the size 3x3 with 3x6 with rank 40.
