I'm currently reading the following paper on persistent homology: https://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf.
Given a filtration of a simplicial complex, $K$,
$$\{0\}=K^0\subseteq K^1\subseteq,...,\subseteq K^n=K$$ we have the following commutative diagram 
where the chain $\{C_*^i, \partial_*^i\}$ corresponds to the chain complex of the simplicial complex of $K^i$. The chain homomorphisms $f^i$ correspond to the inclusion maps induced as a result of the filtration i.e. a $d$-simplex in $K^i$ is a $d$-simplex in $K^{i+1}$.
As a result, we have the following persistence module, for each dimension $d$: $$H_d^0\rightarrow H_d^1\rightarrow H_d^2\rightarrow...\rightarrow H_d^n$$ where arrows are just the linear maps induced by the inclusion maps $f^i$.
We can take the direct sum of these homology groups to give a graded $F[t]$ module which according to structure theorem for graded modules is isomorphic to:
Using this decomposition one can produce a barcode of the homology groups. However, in the linked paper in section 4, pages 8 and 9, the authors use an algorithm on the matrix represention of the $\partial_{d}^{n+1}$ map in order to calculate all the corresponding barcodes.
I don't see how all the information within the persistence module is encoded with the matrix structure of $\partial_{d+1}^{n}$. Can someone care to explain?
I have been stuck on it for weeks and still do not understand why that should be the case.