Question: Standard algorithms have been proposed to compute the partial derivative of eigenvalue and eigenvector w.r.t. the matrix, e.g. <http://www.win.tue.nl/casa/meetings/seminar/previous/_abstract051019_files/Presentation.pdf>.

However, as far as I know, no solution has derived for the partial derivative of the matrix w.r.t. its eigenvalue and eigenvector. Suppose the matrix $A\in \mathbb{R}^{n \times n}$ with eigenvalue $\lambda\in \mathbb{R}$ and eigenvector $X\in \mathbb{R}^{n}$, i.e. $ AX = \lambda X$,the problem is to compute $\frac{\partial A}{\partial \lambda}$ and $\frac{\partial A}{\partial X}$. Does anyone have some ideas? Thanks for your help!

Answer: 
@ bbl , you mix all these concepts. Firstly, an eigenvalue or a unitary eigenvector of $A=[a\_{i,j}]$ is a function of the $a\_{i,j}$ 

$f:A \rightarrow \lambda$ and $g:A\rightarrow x\_{\lambda}$

and $Df\_A$ or $Dg\_A$ (when they exist !) are the derivatives of these functions and absolutely not PARTIAL derivatives.

On the other hand, when $A$ is diagonalizable, then (cf. Victor's post) $A$ is a function of $\Lambda$ and $X$ (the columns of $X$ are assumed to be unitary and linearly independent ; therefore the couple $(\Lambda,X)$ depends on $n^2$ parameters). Finally, under some hypothesis and precautions, we can consider PARTIAL derivatives with respect to $\Lambda$ or $X$.

Victor gives the first partial derivative in a pretty formula ; I prefer the following form: 

$ \dfrac{\partial A}{\partial \lambda\_i} = X\_i [(X^{-T})\_i]^T$.
$\def\L{\left}\def\R{\right}$Since no one has yet addressed it,
here is the second of the requested gradients.

Using the same assumptions as Victor, calculate the differential wrt $X$
$$\eqalign{
 A &= X\Lambda X^{-1} \\
dA &= dX\Lambda X^{-1} + X\Lambda dX^{-1} \\
 &= dX\Lambda X^{-1} - X\Lambda X^{-1}\,dX\,X^{-1} \\
 &= I\_n\,dX\Lambda X^{-1} - A\,dX\,X^{-1} \\
}$$
Vectorizing both sides of this expression yields the desired gradient
$$\eqalign{
da &= \Big((\Lambda X^{-1})^T\otimes I\_n - X^{-T}\otimes A\Big)\,dx \\
 &= \L(X^{-1}\otimes I\_n\R)^T\L(\Lambda\otimes I\_n-I\_n\otimes A\R)dx \\
\frac{\partial a}{\partial x}
 &= \L(X^{-1}\otimes I\_n\R)^T \L(\Lambda\otimes I\_n - I\_n\otimes A\R) \\
}$$