Matrix Method for Persistence Modules on Commutative Ladders of Finite Type

The theory of persistence modules on the commutative ladders $C{L}_n(\tau )$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module $M$ on $C{L}_n(\tau )$ as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case ($n\le 4)$, we provide an algorithm that uses certain permissible row and column operations to compute a normal form of the block matrix. In this form an indecomposable decomposition of $M$, and thus its persistence diagram, is obtained.