  The notion of a matrix factorization was introduced by Eisenbud in the
commutative case in his study of bounded (periodic) free resolutions over
complete intersections. In this work, we extend the notion of (homogeneous)
matrix factorizations to regular normal elements of connected graded algebras
over a field.
  Next, we relate the category of twisted matrix factorizations to an element
over a ring and certain Zhang twists. We also show that in the AS-regular
setting, every sufficiently high syzygy module is the cokernel of some twisted
matrix factorization. Furthermore, we show that in this setting there is an
equivalence of categories between the homotopy category of twisted matrix
factorizations and the singularity category of the hypersurface, following work
of Orlov.
