For a matrix $A \in \mathbb{R}^{2\times 2}$ let its singular values be $\sigma_+(A) > 0$ and $\sigma_-(A)\geq 0$.
What is the expression for singular values of $M = A +A^\top$, i.e., what are $\sigma_+(M)$ and $\sigma_-(M)$ as functions of $\sigma_+(A)$ and $\sigma_-(A)$?
Is this a hard problem? Initially I though this should be simple, but I struggle to find the expression.
Edited later 1: I'm actually now not sure that $\sigma_+(M)$ and $\sigma_-(M)$ only depend on $\sigma_+(A), \sigma_-(A)$, they might depend on elements of $A$.
Edited later 2: Following the discussion to the answer given by user1551, I think the answer might change if one instead considers signed singular values (which is the definition used in geometric processing. So if you, say, have triangular mesh, then you want to not only measure how much a transformation stretches the space (unsigned, the usual, singular values) but also track if it does flipping of a surface, i.e., changing its orientation (signed singular values)).
You cannot express the singular values of $A+A^T$ in terms of the singular values of $A$. Consider $$ A(t)=e^{tK}=\underbrace{(e^{tK})(I)(I)}_{\text{SVD}} \text{ where } K=\pmatrix{0&-1\\ 1&0}. $$ The singular values of $A(t)$ are all equal to $1$, but the singular values of $A(0)+A(0)^T=2I$ are all equal to $2$ but the singular values of $A(\frac{\pi}{2})+A(\frac{\pi}{2})^T=K+K^T=0$ are zeroes.