The PageRank algorithm is used today within web information retrieval to provide a content-neutral ranking metric over web
pages. It employs power method iterations to solve for the steady-state vector of a DTMC. The defining one-step probability
transition matrix of this DTMC is derived from the hyperlink structure of the web and a model of web surfing behaviour which
accounts for user bookmarks and memorised URLs.
In this paper we look to provide a more accessible, more broadly applicable explanation than has been given in the literature
of how to make PageRank calculation more tractable through removal of the dangling-page matrix. This allows web pages without
outgoing links to be removed before we employ power method iterations. It also allows decomposition of the problem according
to irreducible subcomponents of the original transition matrix. Our explanation also covers a PageRank extension to accommodate
TrustRank. In setting out our alternative explanation, we introduce and apply a general linear algebraic theorem which allows
us to map homogeneous singular linear systems of index one to inhomogeneous non-singular linear systems with a shared solution
vector. As an aside, we show in this paper that irreducibility is not required for PageRank to be well-defined.