the messiest matrix that still admits it's basically diagonal, if you squint and add ones.
means a square matrix with a repeated eigenvalue down the diagonal, ones on the superdiagonal, and zeros everywhere else, used to describe linear maps that can't be fully diagonalized.
from named after camille jordan, the french mathematician who proved in the 1870s that every matrix over the complex numbers is similar to a block-diagonal matrix built from these pieces, now called jordan normal form.
repeated root odes — differential equations with repeated characteristic roots produce t times e to the rt solutions
markov chain analysis — nearly-decoupled states create jordan-like transition structure near eigenvalue 1
control theory poles — repeated poles in a transfer function force jordan block realizations of state matrices