Continuous factorization of the identity matrix

TL;DR

This paper explores the conditions for continuously factorizing the identity matrix using a continuous matrix function, analyzing how dimensions, entries, and norms influence the factorization process.

Contribution

It provides a theoretical framework linking the dimension, entries, and norms of the factorizing matrix to the original identity matrix.

Findings

Identifies conditions for continuous factorization of the identity matrix.

Establishes relationships between matrix dimensions, entries, and norms.

Provides theoretical insights into matrix factorization constraints.

Abstract

We investigate conditions under which the identity matrix $I_n$ can be continuously factorized through a continuous $N\times N$ matrix function $A$ with domain in $\mathbb{R}$. We study the relationship of the dimension $N$, the diagonal entries of $A$, and the norm of $A$ to the dimension $n$ and the norms of the matrices that witness the factorization of $I_n$ through $A$.