Q:

Derivative for eigenvalue with respect to 1st / 2nd / 3rd invariant of a matrix

Definition
There is a 3 by 3 matrix $A$ where $Ax=\lambda x$, so the $\lambda$， where $\lambda$ and $x$ are eigenvalues and eigenvectors of matrix $A$.
And then we have the invariants of the matrix, 
$I_1 = tr(A) = A_{11} + A_{22} + A_{33} = \lambda_1 + \lambda_2 + \lambda_3$
$I_2 = \frac{1}{2}\{[tr(A)]^2-tr(AA)\}=A_{11}A_{22} + A_{22}A_{33} + A_{11}A_{33} - A_{12}A_{21} - A_{23}A_{32}-A_{13}A_{31} = \lambda_1\lambda_2 + \lambda_2\lambda_3 + \lambda_1\lambda_3 $
$I_3 = det(A) = \lambda_1\lambda_2\lambda_3$
Question

Is there a closed form expression for $\frac{\partial \lambda_i}{\partial I_1}$, $\frac{\partial \lambda_i}{\partial I_2}$, and $\frac{\partial \lambda_i}{\partial I_3}$?
Where will this question evolve if A is a symmetric matrix?

P.S: the application of this problem is in solid mechanics, I'd like to know the derivative of the principal stretch with respect to the first/second invariant of the right Cauchy-Green tensor.

A:

If $\boldsymbol{A}$ is a symmetric second-order tensor, its eigenvalues $\lambda_1$, $\lambda_2$, $\lambda_3$ are all real. The eigenvalues are related to the invariants $I_1$, $I_2$, $I_3$ of $\boldsymbol{A}$ through the characteristic polynomial
$$
-\lambda^3 + I_1 \lambda^2 - I_2 \lambda + I_3 = 0 \, ,
$$
of which they are the (real) roots.
The roots of this polynomial can be obtained using Cardano's method. With the change of variable $\lambda = X + \frac{1}{3}I_1$, the previous equation rewrites as
$$
X^3 + \underbrace{\left(I_2 - \frac{{I_1}^2}{3}\right)}_p X + \underbrace{\frac{I_1 I_2}{3} - \frac{2 {I_1}^3}{27} - I_3}_q = 0 \, .
$$
All eigenvalues are real if the discriminant $\Delta = - (4 p^3  + 27 q^2)$ is positive. Using the fact that $\Delta$ is a degree-two polynomial of $I_3$, one obtains the conditions ${I_1}^2 - 3 I_2 > 0$ and
$$
{-1} < \frac{27 I_3 - 9 I_1 I_2 + 2 {I_1}^3}{2 ({I_1}^2 - 3 I_2)^{3/2}} <1 \, ,
$$
to ensure that all eigenvalues $\lambda_1$, $\lambda_2$, $\lambda_3$ are real. Their expression in terms of the invariants $I_1$, $I_2$, $I_3$ can be deduced from the polar decomposition of the Cardano formulas:
$$
\lambda_k = \frac{I_1}{3} + 2\sqrt{\frac{-p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{-q}{2}\sqrt{\frac{27}{-p^3}}\right) + \frac{2k\pi}{3}\right) ,
$$
for $k\in\lbrace 1,2,3\rbrace$.
It remains to compute the derivatives $\partial\lambda_k/\partial I_\ell$.

