Matrix method for persistence modules on commutative ladders of finite type

The theory of persistence modules on the commutative ladders \(CL_n(\tau )\) 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 \(CL_n(\tau )\) 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.