  We investigate the emergent factorization of Hilbert space in the low-energy
description of matrix models, addressing key aspects of the black hole
information paradox. We examine the collective description for the low-energy
sector of $SU(N)$ matrix model, characterized by a factorized Hilbert space
composed of a finite number of boxes and anti-boxes. This factorization leads
us to examine the emergence of thermofield dynamics (TFD) state in the low
energy sector from a fine-tuned state. In addition, we study the collective
Hamiltonian of the $U(N)$ matrix model for the semi-classical description of
"particle-hole" fluctuations around a background Young tableau. Our
investigation of these matrix models elucidates a concrete mechanism for
constructing the truncated algebra of accessible observables, thereby
facilitating an understanding of black hole complementarity. In the context of
the black hole information paradox, we discuss the origin of the island
appearing inside the black hole and provide a reinterpretation of the recent
proposal -- the holography of information.
