Parametric Factorization of Matrices

TL;DR

This paper surveys parametric matrix factorizations in complex groups, focusing on algebraic, continuous, and holomorphic cases, and introduces new bounds for specific matrix factorizations.

Contribution

It provides a comprehensive account of holomorphic and algebraic matrix factorizations, including a new lower bound for $SL_2(\mathbb{C})$ matrices parametrized by higher-dimensional spaces.

Findings

Established a new lower bound for $SL_2(\mathbb{C})$ matrix factorizations.

Reviewed algebraic, continuous, and holomorphic factorization results.

Focused on holomorphic factorization in the context of Several Complex Variables.

Abstract

In this survey paper we study parametric versions of writing a matrix in $SL_n (\mathbb{C})$ as a product of lower and upper unitriangular matrices in interchanging order as well as generalizations to other classical groups. We give an account of algebraic, continuous and holomorphic factorization results, from the standpoint of Several Complex Variables. Out of the wealth of algebraic results, we only concentrate on those which are related to holomorphic factorization and often formulate them in a specific form, e.g. for the field of complex numbers in place of more general fields or principal ideal domains. The number of unitriangular matrices needed is a difficult problem and is solved in very specific cases only. We give a new lower bound for factorizing matrices in $SL_2 (\mathbb{C})$ continuously parametrized by normal topological spaces of dimension bigger than one.