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.
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.