  The spectral statistics of non-Hermitian random matrices are of importance as
a diagnostic tool for chaotic behavior in open quantum systems. Here, we
investigate the statistical properties of singular values in non-Hermitian
random matrices as an effective measure of quantifying dissipative quantum
chaos. By means of Hermitization, we reveal the unique characteristics of the
singular-value statistics that distinguish them from the complex-eigenvalue
statistics, and establish the comprehensive classification of the
singular-value statistics for all the 38-fold symmetry classes of non-Hermitian
random matrices. We also analytically derive the singular-value statistics of
small random matrices, which well describe those of large random matrices in
the similar spirit to the Wigner surmise. Furthermore, we demonstrate that
singular values of open quantum many-body systems follow the random-matrix
statistics, thereby identifying chaos and nonintegrability in open quantum
systems. Our work elucidates that the singular-value statistics serve as a
clear indicator of symmetry and lay a foundation for statistical physics of
open quantum systems.
