With the elliptical gradient-index medium used as the optical system to implement two-dimensional axial symmetric fractional Fourier transforms (FRFT),the analytical formula of the beam width and the intensity distribution for two-dimensional axial symmetric fractional Fourier transform of elliptical Gaussian beams have been derived,based on the equivalence between the rotation of the Wigner distribution function and the fractional Fourier transform. By employing the derived formula,the influence of the fractional order on the properties of elliptical Gaussian beams in the fractional Fourier transform plane has been studied in detail. The results indicate that the on-axis intensity and the beam width of elliptical Gaussian beams change asynchronously along the x,y axises and the two parameters are dependent on the fractional order px and py respectively. The circular Gaussian beams will be obtained in specific output plane. Meanwhile,it will appear that the beam width satisfies wx>wy in specific output plane in spite of wx