  Matrices with off-diagonal decay appear in a variety of fields in mathematics
and in numerous applications, such as signal processing, statistics,
communications engineering, condensed matter physics, and quantum chemistry.
Numerical algorithms dealing with such matrices often take advantage
(implicitly or explicitly) of the empirical observation that this off-diagonal
decay property seems to be preserved when computing various useful matrix
factorizations, such as the Cholesky factorization or the QR-factorization.
There is a fairly extensive theory describing when the inverse of a matrix
inherits the localization properties of the original matrix. Yet, except for
the special case of band matrices, surprisingly very little theory exists that
would establish similar results for matrix factorizations. We will derive a
comprehensive framework to rigorously answer the question when and under which
conditions the matrix factors inherit the localization of the original matrix
for such fundamental matrix factorizations as the LU-, QR-, Cholesky, and Polar
factorization.
