Given a matrix $\mathbf{X} \in \mathbb{R}^{N \times N}$ and some function $\phi : \mathbb{R}^{N \times N} \to \mathbb{R}^N$, I have tried to understand how the function value is affected when a small perturbation $\mathbf{\epsilon}$ added to the matrix.
Can I approximate $\phi(\mathbf{X}+\mathbf{\epsilon})$ using $\phi(\mathbf{X})$ and $\mathbf{\epsilon}$?
I have thought some possible solutions such as the first order approximation like this. However, I don't how accommodate the solutions to my problem since $\phi$ maps matrix to vector.
Can anybody help me with this problem? Any answer is appreciated.
If the gradient of the function (a third-order tensor) is calculated as
$${\cal G} = \frac{\partial \phi}{\partial X} \quad\implies {\cal G}_{ijk} = \frac{\partial \phi_{i}}{\partial X_{jk}}$$ Then the first two terms of the Taylor expansion are
$$\eqalign{ \phi(X+E) &= \phi(X) + {\cal G}:E \\ }$$ The double-dot product is more clearly expressed using index notation $$\eqalign{ \big({\cal G}:E\big)_i &= \sum_{j}\sum_{k}{\cal G}_{ijk}\,E_{jk} \\ }$$