  When a matrix has a banded inverse there is a remarkable formula that quickly
computes that inverse, using only local information in the original matrix.
This local inverse formula holds more generally, for matrices with sparsity
patterns that are examples of chordal graphs or perfect eliminators. The
formula has a long history going back at least as far as the completion problem
for covariance matrices with missing data. Maximum entropy estimates,
log-determinants, rank conditions, the Nullity Theorem and wavelets are all
closely related, and the formula has found wide applications in machine
learning and graphical models. We describe that local inverse and explain how
it can be understood as a matrix factorization.
