Let $A\in \mathcal{M}_n(\mathbb{R})$ be a diagonal positive matrix. We assume that $A$ is generic (in a sense to clarify). Let $\lambda \in\mathbb{R}$ and $U\in O_n(\mathbb{R})$ ($UU^T=I$) be such that $A+\lambda U$ has a multiple singular value (that is $A²+\lambda(UA+AU^T)$ has a double eigenvalue). Q1. Has $U$ necessarily an entry equal to $\pm 1$ on its diagonal ? Q2. Do there exist only a finite number of corresponding values of $\lambda$ ? Thanks in advance.

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noneof the matrices $M+xN$ (for $x \in \mathbb R$) has double eigenvalues. This is a surprising phenomenon known aseigenvalue avoidance. References: Lax'sLinear Algebra and its Applications(the book, not the journal with the same name) and eprints.maths.ox.ac.uk/1175. – Federico Poloni Jun 21 '13 at 19:17