Hm, yes, if I understood the question correctly, then yes I am working with matrices and vectors restricted to reals. About change of basis, I'm not quite sure about that, I have learned how to make orthonormal basis vectors out of linearly independent vectors though (Gram-Schmidt method)...
Let's say I have a matrix M such that for vectors R and r in xy-coordinate system:
R=Mr
Suppose we diagonalized it so that there is another matrix D such that for vectors R' (which is also R) and r' (which is also r) in x'y'-coordinate system:
R'=Dr'
D is a matrix with zero elements except for...