Principle Axes of inertia and moments of inertia 
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*Principal axes of inertia is defined as the eigenvectors of the inertia tensor matrices. I understand that a diagonalised tensor can yield these axes, but why are they necessarily form the axes of a rotating frame? Can anyone explain why?

*Lastly, why is it true that any symmetric axis is always a principal axis? 

*If $I_{xx} = I_{yy}$,  why is any linear combination of two orthogonal directions on the place spanned by the eigenvector $e_1$ and $e_2$ can always be chosen as principal axes?
I'm really confused... Hoping that anyone could shed a light for me.
 A: 1) I don't understand your questions really well. Consider rephrasing it to make it a bit clearer. 
What I can tell you is the following: 
The mathematical definition of the principal axes of inertia is that they are the eigenvectors of the inertia tensor. 
The physical interpretation of the principal axes of inertia is that they represent the directions along which the angular momentum $\mathbf{L}$ is parallel to the angular velocity $\mathbf{\omega}$, so $\mathbf{L} = I\mathbf{\omega} $ with $I$ = constant as opposed to $\mathbf{L} = \underline{\underline{I}} \cdot \mathbf{\omega} $. 
When calculating the inertia tensor $\underline{\underline{I}} $, you use a certain basis (so that you find all the distances between the parts of the system). If that basis were to be the set of principal axes, then the matrix would come out to be diagonal. In other words, if you want to diagonalise the inertia tensor via a similarity transformation $\underline{\underline{S}}^{-1}\underline{\underline{I}}\underline{\underline{S}}$, you would have to use the matrix made of the principal axes of inertia as $\underline{\underline{S}}$.
2) There are some tricks to determine which axes are principal axes. They can be shown mathematically but usually a qualitative understanding is enough (i.e. it makes sense - why would they not be?).
If an object has a rotational symmetry (i.e. you can rotate about an axis without changing anything about it - shape, mass distribution etc.) then the axis of symmetry is a principal axis. 
If an object has a reflection/mirror symmetry (i.e. an equilateral triangle is symmetric about a line/plane through a vertex and the mid-point of the opposite edge), then the direction normal to the plane of symmetry is a principal axis.
3) In the case of a diagonal inertia tensor (i.e. using the principal axes as the basis), $I_{xx} = I_{yy}$ means that two eigenvalues are the same and therefore they are degenerate. If you try and calculate the eigenvector associated with this eigenvalue you will find a plane (an equation in terms of x, y and z). Basically any vector on that plane will be a principal axis, but it is customary to choose two orthonormal vectors, just to make life easier.
