Continuous factorization of the identity matrix

Faculty Member

Pavlos Motakis

We say that an NxN matrix A is a factor of the nxn identity matrix I if there are an nxN matrix L and an Nxn matrix R so that LAR = I. If A is an NxN matrix so that

1. N is large enough compared to n and

2. the absolute values of the diagonal entries of A divided by the operator norm of A are above a certain threshold

then A is always a factor of the nxn identity matrix, with some additional control on the operator norms of matrices L and R.

We may replace A, L, and R with continuous matrix valued functions depending on a real variable t to obtain the definition of a continuous factor of the identity. The purpose of the project is to investigate whether a condition on the diagonal entries of the matrix function A can imply that A is a continuous factor of the identity.

The first stage of the project consists of giving a rigorous proof of the first statement while giving concrete numerical relations between n, N, and the operator norms of the matrices involved. This involves making a connection between orthonormal vectors in a Euclidean space and combinatorial principles.

The second and main stage will be trying to overcome the challenge of generalizing this from a matrix to a continuous matrix function.

Team Meetings

Biweekly

Project Difficulty

Intermediate

Undergrad Prerequisites

Completion of Math 416 (Abstract Linear Algebra) and Math 447 (Real Variables) are required. Completion of Math 412 (Graph Theory) is recommended but not necessary.