Explicit formulas concerning eigenvectors of weakly non-unitary matrices

Abstract

We investigate eigenvector statistics of the Truncated Unitary ensemble $\mathrm{TUE}(N,M)$ in the weakly non-unitary case $M=1$, that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when $M\ge N$, suggesting possible extensions to $\mathrm{TUE}(N,M)$ for all values of $M$.

11 pages, 1 figure

References in corpus (2)

• Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

Cited by in corpus (3)

• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• Dynamics of a rank-one multiplicative perturbation of a unitary matrix