Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda\_1,\cdots,\lambda\_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall denote the eigenvalues as $\lambda\_i(A)$. 


**Question** What can be said about the differentiability of the functions $\lambda\_i:GL(n,\mathbb{R}) \to \mathbb{R}$? 


[*We may assume that the domain is $GL^+(n,\mathbb{R})$ for differentiability/smoothness.*]


Any reference for this or relevant results would be appreciated.



Differentiability of eigenvalues of positive-definite symmetric matrices
In the open subset of $M\_n(\mathbb{R})$ where the $\lambda\_i$ are distinct, they are $C^{\infty}$ functions: this follows from the implicit function theorem.


On the other hand, when some eigenvalue has multiplicity $>1$ you don't get more than continuity. For example if $A=\begin{pmatrix} 0 & 1\\ 1 & t
\end{pmatrix}$ the largest $\lambda\_i$ is $\dfrac{1}{2}\left(t^2+2 +|t|\sqrt{t^2+4}\right)$, which is not differentiable (as a function of $t$) at $t=0$.


