The linear dependence problem for power linear maps Original Research Article

Let Bl, l = 1, … , k, be m × nl complex matrices and let image be complex vector variables. We show that the components of the map H=(B1x[1])(d1)ring operatorcdots, three dots, centeredring operator(Bkx[k])(dk) are linearly dependent over image if and only if image, where ring operator means the Hadamard product, X* and X(d) denote the conjugate transpose and the dth Hadamard power of a matrix X respectively. Connections are established between the Homogenous Dependence Problem (HDP(n,d)), which arises in the study of the Jacobian Conjecture, and the dependence problem for power linear maps (PLDP(n,d)). An algorithm is given to compute counterexamples to PLDP(n,d) from those to HDP(n,d), and counterexamples to PLDP(n,3) are obtained for all ngreater-or-equal, slanted67.