  On the case that the number of dangling nodes is large, PageRank computation
can be proceeded with a much smaller matrix through lumping all dangling nodes
of a web graph into a single node. Thus, it saves many computational cost and
operations. There are also some theoretical contributions on Jordan canonical
form of the Google matrix. Motivated by these theoretical contributions, in
this note, we provide alternative proofs for some results of Google matrix
through the lumping method due to Ipsen and Selee. Specifically we find that
the result is also suitable for some subsequent work based on lumping dangling
nodes into a node. Besides, an entirely new proof from the matrix decomposition
viewpoint is also proposed.
