If complex numbers are allowed, then there always exists a transformation matrix between two metric tensors. So I guess complex numbers are not allowed.
Could you explain me why?
The components of transformation matrix can be complex number, can't they?
Here, I am looking for the transformation matrix between two coordinate systems while the components of metric tensor are given for the two coordinate systems.
So, there are ##\frac{N+N^2}{2}## equations and ##N^2## unknowns (the ##N^2## components of ##\frac{\partial \chi^i}{\partial x^j}##). As ##N^2>\frac{N+N^2}{2}##, there will be infinite number of solutions.
Sorry for the mistake.
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