  We introduce one-dimensional lattice models with exact matrix-product ground
states describing the fractional quantum Hall (FQH) states in Laughlin series
(given by filling factors $\nu=1/q$) on torus geometry. Surprisingly, the
exactly solvable Hamiltonian has the same mathematical structure as that of the
pseudopotential for the Laughlin wave function, and naturally derives the
general properties of the Laughlin wave function such as the $Z_2$ properties
of the FQH states and the fermion-boson relation. The obtained exact ground
states have high overlaps with the Laughlin states and well describe their
properties. Using the matrix product method, density functions and correlation
functions are calculated analytically. Especially, obtained entanglement
spectra reflects gapless edge states as was discussed by Li and Haldane.
