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.