We measure the matter power spectrum from 31 Ly \alpha  spectra spanning the redshift range of 1.6-3.6 .
The optical depth , \tau , for Ly \alpha  absorption of the intergalactic medium is obtained from the flux using the inversion method of Nusser & Haehnelt ( 1999 ) .
The optical depth is converted to density by using a simple power law relation , \tau \propto ( 1 + \delta ) ^ { \alpha } .
The non-linear 1D power spectrum of the gas density is then inferred with a method that makes simultaneous use of the 1 and 2 point statistics of the flux and compared against theoretical models with a likelihood analysis .
A Cold Dark Matter ( CDM ) model with standard cosmological parameters fits the data well .
The power spectrum amplitude is measured to be ( assuming a flat Universe ) , \sigma _ { 8 } = ( 0.9 \pm 0.09 ) \times ( \Omega _ { m } / 0.3 ) ^ { -0.3 } , with \alpha varying in the range of 1.56 - 1.8 with redshift .
Enforcing the same cosmological parameters in all four redshift bins , the likelihood analysis suggests some evolution in the density-temperature relation and the thermal smoothing length of the gas .
The inferred evolution is consistent with that expected if reionization of He ii  occurred at z \sim 3.2 .
A joint analysis with the WMAP results together with a prior on the Hubble constant as suggested by the HST key project data , yields values of \Omega _ { m } and \sigma _ { 8 } that are consistent with the cosmological concordance model .
We also perform a further inversion to obtain the linear 3D power spectrum of the matter density fluctuations .