Complex Eigenvalues in a pseudo-Hermitian \b{eta}-Laguerre ensemble

Non-Hermitian PT-symmetric models have been extensively studied in recent years. Following the seminal work that reduced classical random matrix ensembles to a tridiagonal form, several efforts have aimed to generalize this framework to non-Hermitian extensions of the so-called \b{eta}-ensembles. In particular, while the transition of eigenvalues from the real axis to the complex plane has been well characterized for the \b{eta}-Hermite ensemble under symmetry breaking, the behavior of the \b{eta}-Laguerre ensemble in a similar non-Hermitian setting remains less understood. In this work, we investigate an ensemble of unstable matrices isospectral to the \b{eta}-Laguerre ensemble. Introducing a small non-Hermitian perturbation breaks the symmetry and drives the eigenvalues into the complex plane. We derive analytical expressions for the loci of complex-conjugate eigenvalue pairs, which organize into a balloon-like structure in the complex plane, followed by a discrete finite line of real eigenvalues. The asymptotic behavior of these eigenvalues is analyzed in the large matrix-size limit, and our theoretical predictions are supported by numerical simulations.