Given an invertible matrix $A \in \mathbb{R}^{n \times n}$ and its polar decomposition $$ A = U H, $$ where $U$ is unitary, and $H$ is positive definite. I'm interested in the distance of $$ arg ( \lambda_A ) - arg( \lambda_U ) $$ where $arg ( \lambda_A ) $ denotes the arguement (phase) of the eigenvalues of $A$ in an appropriate order. For normal matrices it is that $arg ( \lambda_A ) - arg( \lambda_U ) = 0$.
Is there a way to estimate the angular difference for non-normal matrices?