  Given a real $n \times m$ matrix $B$, its operator norm can be defined as
$$|B|=\max_{|v|=1}|Bv|.$$ We consider a matrix "small" if it has non-negative
integer entries and its operator norm is less than $2$. These matrices
correspond to bipartite graphs with spectral radius less than $2$, which can be
classified as disjoint unions of Coxeter graphs. This gives a direct route to
an $ADE$-classification result in terms of very basic mathematical objects. Our
goal here is to see these results as part of a general program of
classification of small objects, relating quadratic forms, reflection groups,
root systems, and Lie algebras.
