In this paper, we view matrix frames as representations of quivers and study them within the general framework of quiver invariant theory. We are thus led to consider the large class of semi-stable matrix frames. Within this class, we are particularly interested in radial isotropic and Parseval matrix frames.Using methods from quiver invariant theory [CD19], we first prove a far reaching generalization of Barthe's Radial Isotropy Theorem [Bar98] to matrix frames (see Theorems 1(3) and 28). With this tool at our disposal, we provide a quiver invariant theoretic approach to Paulsen's problem for matrix frames. We show in Theorem 2 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 (F , W) ≤ 26ǫd 2 . CONTENTS 1. Introduction 1 2. Background on Quiver Invariant Theory 6 3. Matrix Radial Isotropy 11 4. Paulsen Problem for matrix frames 13 5. Constructive aspects of Matrix Radial Isotropy 18 6. Quiver Radial Isotropy and σ-critical quiver representations 21 References 25

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