Matrix Factorization and Lifting

Abstract

As a result recent interdisciplinary work, the processing of signals (audio, still-images, etc), a number of powerful matrix operations have led to advances both in engineering applications, and in mathematics. Much of it is motivated by ideas from wavelet algorithms. The applications are convincing measured against other processing tools already available, for example better compression (details below). We develop a versatile theory of factorization for matrix functions. By a matrix valued function we mean a function of one or more complex variables taking values in the group GL(n, C) of invertible n by n matrices. Starting with this generality, there is a variety of special cases, also of interest, for example, one variable, or restriction to the case n = 2; or consideration of subgroups of GL(n, C); or SL(n, C), i.e., specializing to the case of determinant equal to one. We will prove a number of factorization theorems and sketch their application to signal (image processing) in the framework of multiple frequency bands.