  Starting from a Hermitian operator with two distinct eigenvalues, we
construct a non-Hermitian bipartite system in Gaussian orthogonal ensemble
according to random matrix theory, where we introduce the off-diagonal
fluctuations through random eigenkets and realizing the bipartite configuration
consisting of two $D\times D$ subsystems (with $D$ the Hilbert space
dimension). As required by the global thermalization (chaos), one of the two
subsystems is full ranked, while the other is rank deficient. For the latter
subsystem, there is a block with non-defective degeneracies containing the
non-linear symmetries, as well as the accumulation effect of the linear map in
adjacent eigenvectors. The maximally mixed state made by the eigenvectors of
this special region exhibit not thermal ensmeble behavior (neither canonical or
Gibbs), and exhibit similar character with the corresponding reduced density,
which can be verified through the Loschmitch echo and variance of the imaginary
spectrum. This non-defective degeneracy region partly meets the Lemma in
10.1103/PhysRevLett.122.220603 and theorem in 10.1103/PhysRevLett.120.150603.
The coexistence of strong entanglement and initial state fidelity in this
region make it possible to achieve a maximally mixed density which, however,
not be a thermal canonical ensemble (with complete insensitivity to the
environmental energy or temperature). Outside this region, the collection of
eigenstates (reduced density) always exhibit restriction on the corresponding
Hilbert space dimension, and thus suppress the thermaliation. There are
abundant physics for those densities in Hermitian and non-Hermitian bases,
where we investigate seperately in this work.
