We report the transverse phase space diagnostics for electron beams generated through ionization injection in a laser-plasma accelerator. Single-shot measurements of both ultimate emittance and Twiss parameters are achieved by means of permanent magnetic quadrupole. Beams with emittance of μm rad level are obtained in a typical ionization injection scheme, and the dependence on nitrogen concentration and charge density is studied experimentally and confirmed by simulations. A key feature of the transverse phase space, matched beams with Twiss parameter αT ≃ 0, is identified according to the measurement. Numerical simulations that are in qualitative agreement with the experimental results reveal that a sufficient phase mixing induced by an overlong injection length leads to the matched phase space distribution.