We present a thorough study of the theoretical properties and devise efficient algorithms for the \emph{persistent Laplacian}, an extension of the standard combinatorial Laplacian to the setting of pairs (or, in more generality, sequences) of 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 the $q$-th persistent Laplacian $\Delta_q^{K,L}$ equals the $q$-th persistent Betti number of the inclusion $(K \hookrightarrow L)$. We then present an initial algorithm for finding a matrix representation of $\Delta_q^{K,L}$, which itself helps interpret the persistent Laplacian. We exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix which has several important implications. In the graph case, it both uncovers a link with the notion of effective resistance and leads to a persistent version of the Cheeger inequality. This relationship also yields an additional, very simple algorithm for finding (a matrix representation of) the $q$-th persistent Laplacian which in turn leads to a novel and fundamentally different algorithm for computing the $q$-th persistent Betti number for a pair $(K,L)$ which can be significantly more efficient than standard algorithms. Finally, we study persistent Laplacians for simplicial filtrations and present novel stability results for their eigenvalues. Our work brings methods from spectral graph theory, circuit theory, and persistent homology together with a topological view of the combinatorial Laplacian on simplicial complexes.

翻译：我们展示了对理论属性的彻底研究,并设计了用于\ emph{ persistant Laplacian} 的高效算法, 标准组合式拉普拉西亚(Laplacian) 的纯值等于 $ (K\ hookrightrow L$ ) 的固定值。 我们随后展示了一种初步的算法, 用于寻找 $\ Delta_ qqquarrowL$ 的矩阵代表, 由Wang、 Nguyen和Wei最近推出的。 特别是, 与非持久性案例相比, 我们首先证明, 美元不变的平价的平价计算值等于 美元 美元 的固定值 。 在图表中, 美元 美元 美元 美元 的固定值等于 美元 的固定值等于 美元, 以 美元 基数 的基数 的基数 。 一种基数的基数 和 美元 基数的基数 的基数是 。