  We propose and prove two theorems for determining the number of dark modes in
linear two-component quantum networks composed of two types of bosonic modes.
This is achieved by diagonalizing the two sub-networks of the same type of
modes, mapping the networks to either a standard or a thick arrowhead matrix,
and analyzing the linear dependence and independence between the column vectors
associated with degenerate normal modes in the coupling matrix. We confirm the
two theorems by checking the simultaneous ground-state cooling of the
mechanical modes in linearized optomechanical networks. These results also work
for linear fermionic networks and other networks described by quadratic
coupled-mode Hamiltonian. The present method can be extended to study the
dark-state effect in driven atom systems and to construct large
decoherence-free subspaces for processing quantum information. This work will
initiate the studies on dynamical, transport, and statistical properties of
linear networks with decoupled subspaces.
