  We propose a new approach to extract the important degrees of freedom in
quantum dynamics induced by an external stimulus. We calculate the coefficient
matrix numerically, where the $i-l$ element of the matrix is the coefficient of
the lth basis state at the ith discretized time in the solution of the
time-dependent Schr\"odinger equation induced by the external stimulus. By
performing a randomized singular value decomposition of the coefficient matrix,
a practically exact solution is obtained using a linear combination of the
important modes, where the number of modes is much smaller than the dimensions
of the Hilbert space in many cases. We apply this method to analysis of the
light absorption spectrum in two-dimensional (2D) Mott insulators using an
effective model of the 2D Hubbard model in the strong interaction case. From
the dynamics induced by an ultrashort weak light pulse, we find that the
practically exact light absorption spectrum can be reproduced by as few as 1000
energy eigenstates in the $1.7 \times 10^{7}$-dimension Hilbert space of a
26-site cluster. These one-photon active energy eigenstates are classified into
free holon and doublon (H-D) and localized H-D states. In the free H-D states,
the main effect of the spin degrees of freedom on the transfer of a holon (H)
and a doublon (D) is the phase shift, and the H and the D move freely. In the
localized H-D states, an H and a D are localized with relative distances of
$\sqrt{5}$ or $\sqrt{13}$. The antiferromagnetic (AF) spin orders in the
localized H-D states are much stronger than those in the free H-D states, and
the charge localization is of magnetic origin. There are sharp peaks caused by
excitations to the localized H-D states below the broad band caused by
excitations to the free H-D states in the light absorption spectrum.
