  Firstly, we provide a different proof of an important lemma in Buzzard and
Calegari's work on slopes of overconvergent 2-adic modular forms via
nonarchimedean linear Hodge-Newton decomposition. The lemma shows that two
equivalent matrices with coefficients in the ring of integers in an archimedean
field have the same Newton polygon under suitable conditions. Secondly, we give
an archimedean analogue of the above lemma.
