Before revision:  Taking this topological view, Wang et al. introduced the so-called persistent Laplacian \Delta_q^{K,L simplicial complexes K \hookrightarrow L . In this paper, we present a thorough study of properties and algorithms for persistent Laplacians. 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:  Taking this topological view, Wang et al. introduced the so-called persistent Laplacian \Delta_q^{K,L simplicial complexes K \hookrightarrow L , which was recently introduced by Wang, Nguyen, and Wei. In particular, in analogy with the non-persistent case, 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.