A Quiver Invariant Theoretic Approach to Radial Isotropy and the Paulsen Problem for Matrix Frames

Abstract

In this dissertation, we view matrix frames as representations of quivers and study them within the general framework of Quiver Invariant Theory. We are particularly interested in radial isotropic and Parseval matrix frames. Using methods from Quiver Invariant Theory [CD21], we first prove a far-reaching generalization of Barthe's Theorem [Bar98] on vectors in radial isotropic position to the case of matrix frames (see Theorems 5.13(3) and 4.12). With this tool at our disposal, we generalize the Paulsen problem from frames (of vectors) to frames of matrices of arbitrary rank and size extending Hamilton-Moitra's upper bound [HM18]. Specifically, we show in Theorem 5.20 that for any given ε-nearly equal-norm Parseval frame F of n matrices with d rows there exists an equal-norm Parseval frame W of n matrices with d rows such that dist 2 pF, Wq ď 46εd 2 . Finally, in Theorem 5.28 we address the constructive aspects of transforming a matrix frame into radial isotropic position which extend those in [Bar98, AKS20]. v