What is the relation between eigen values and principal axes length for 3D data? For a region in an binary image, I have calculated co-variance matrix using co-ordinates of region. Using co-variance matrix, I got two eigen values. Later I have calculated major axis length and minor axis length using the formula $4\sqrt{\lambda_i}$ where the $\lambda_i$ are the eigenvalues. This formula is mentioned in the below given link  3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix. I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching for 2D object.
Now, I want to calculate principal axes lengths (major, middle and minor axis lengths) for 3D object data. For that I want a relation between eigen values obtained from 3D object co-ordinates (using same procedure mentioned above) and principal axes lengths for 3D data. 'regionprops' command in MATLAB do not work for 3D object.

If I understand correctly your question, your formula seems to be wrong.
In three dimensions the distribution of mass could be represented by the inertia ellipsoid, whose equation, in a principal frame is
$$
I_x x^2 + I_y y^2 + I_z z^2 = 1,
$$
so that the length of the axes of the ellipsoid are $a_i = 1/\sqrt{I_i} = 1/\sqrt{\lambda_i}$ (the principal moments of inertia are the eigenvalues of the inertia matrix).
See section 4.5 of http://www.eng.auburn.edu/~marghitu/MECH2110/C_4.pdf .
