$\mathbf{C}$ is a real, positive-definitive 3x3 symmetric matrix (I am thinking about the right Cauchy-Green tensor in solid mechanics). We perform eigendecomposition and get:

$$\mathbf{C} = \sum_{i=1}^3 \lambda_i^2 \mathbf{N_i} \otimes \mathbf{N_i}$$

where $\lambda_i^2$ are its eigenvalues and $\mathbf{N_i}$ are its eigenvectors.

## My question is, is there a closed form solution for $\frac{\partial \mathbf{N_i}}{\partial \mathbf{C}}$?

PS: I mostly care about in Cartesian coordinate system, where the definition of the derivative of two tensors are: $$(\frac{\partial \mathbf{A}}{\partial \mathbf{B}})_{ijkl} = \frac{\partial A_{ij}}{\partial B_{kl}}$$