Before revision:  We first prove that the nullity of  \Delta_q^{K,L} gives rise to the q-th persistent Betti number from K to L . We then present a first algorithm to compute the matrix representation of \Delta_q^{K,L}, which helps reveal insights into the meaning of persistent Laplacian.
After revision:  We first prove that the nullity of \Delta_q^{K,L} gives rise to the q-th persistent Betti number of the inclusion (K \hookrightarrow L) . We then present a first algorithm to compute the matrix representation of \Delta_q^{K,L}, which helps reveal insights into the meaning of persistent Laplacian.