📂Topological Data Analysis

Matrix Boundaries in Computational Topology

Definition ¹

Given a simplicial complex $K$, let us denote the number of $p$-simplices as $n_{p}$ and the number of $P-1$-simplices as $n_{p-1}$. Assume that the $p-1$th $i$-simplex of dimension $(p-1)$ is a face $a_{i}^{j} = 1$ of the $j$th $p$-simplex and otherwise is $a_{i}^{j} = 0$. The matrix $\partial_{p} := \left[ a_{i}^{j} \right]_{i = 1 , \cdots , n_{p-1}}^{j = 1 , \cdots , n_{p}}$ is called the Boundary Matrix for the $p$nd simplicial complex in $K$.