Explicit formulas concerning eigenvectors of weakly non-unitary matrices

Abstract

We investigate eigenvector statistics of the Truncated Unitary ensemble 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 ≥ N , suggesting possible extensions to TUE(N, M ) for all values of M .

Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities

“…The statistics of non-orthogonality factors are then expected to be exactly the same as given by ( 6), with the perfect coupling case g = 1 corresponding in that context to the so-called truncated CUE [80]. Note that recently non-orthogonality in such ensemble has been rigorously addressed in [91].…”

Rank $1$ perturbations in random matrix theory -- a review of exact results

“…Moreover, this quantity was also calculated for the GOE analogue of (4.1). In another recent development, the overlaps O nm ,, together with certain generalisations referred to as q-overlaps, were studied for the multiplicative sub-unitary rank 1 CUE perturbation of subsection 4.2 in the case a = 0 by Dubach [27].…”

“…Thus the left and right eigenvectors can be chosen so that L i |R j = δ i,j , which in words says that they form a bi-orthogonal family. Note that with respect to this condition, |R j can be multiplied by the scalar α provided |L j is multiplied by 1/ ᾱ. Invariant under such scaling is the so-called overlaps O mn = L m |L n R m |R n , with the diagonal overlaps O nn = ||L n || 2 ||R n || 2 of particular importance for their role as squared eigenvalue condition numbers; see [27] and references therein. The overlaps O mn were studied in the context of the additive rank 1 anti-Hermitian GUE perturbation (4.1) by Fyodorov and Mehlig [53].…”