So the core of the question is the study of the function $$ s :M_n(\mathbb R) \mapsto M_n(\mathbb R)$$ $$A \rightarrow s_n(A) $$
where $s_n(A)$ is the greatest singular value of A. I know there has been a lot of work on this problem (well usually it's about the eigenvalues of symmetric/hermitian matrices but it's related of course).
Now, here the things I know :
if $A^TA$ has simple spectrum, then s has all directionnal derivatives at $A$, and in a linear way (e.g it is Gateaux differentiable). It seems to me it is actually differentiable by the inverse function theorem ?
It is usually nondifferentiable when $A^TA$ has non simple spectrum
Is it true that it is locally-Lipschitz for instance ?
Actually, here what I'm trying to do :
take $A$ and $B$ two matrices such that their symmetrized have simple spectrums. Then one can prove that putting $A_t = A +t(B-A), t \in [0,1]$ there is only a finite number of values of t for which $A_t$ has nonsimple spectrum (algebraic argument), say $t_1, .., t_{n-1}$ and put $t_0 = 0$ and $t_n = n$
Do we have $s_n(B) - s_n(A) = \sum_{i=0}^{n-1} \int_{t_i}^{t_{i-1}}\frac{d}{dt}s_n(A_t) $ ?
It seems to me that the $s(t) = s_n(A_t)$ is differentiable almost everywhere on $[0,1]$ and Lipschitz.
