Let $A$ and $B$ square real matrices. I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1. Can we say something about the eigenvalues of $exp(i\theta)A+B$ with $0\leq \theta < 2\pi$ ? Thank you for your help !
You can't say much. For example, let $A$ be any real diagonal matrix, and then pick a real diagonal matrix $B$ such that $A+B$ satisfies your property. Then $uA+B=(u1)A+(A+B)$, and so the triangle inequality gives $$ u1\A\_21 \leq\uA+B\_2 \lequ1\A\_2+1. $$ That is, the largest eigenvalue of $uA+B$ is on the order of the (unknown) largest eigenvalue of $A$.
Let $C=A+B$ be any real matrix. Let $A$ be any real matrix, then $B=CA$ is a real matrix. Then $e^{i\theta} A+ B = (e^{i\theta}1) A + C$. Since $A$ can be any real matrix, this bears essentially no relationship to $C$!
About the only thing we can say is that the difference of the traces of $e^{i\theta} A +B$ and $A+B$ is $(1e^{i\theta})$ times a real number. Since the trace of $A+B$ is a real number between $d$ and $2d$, with $d$ the dimension of the matrices, this gives you a band of possible values for the trace of $e^{i\theta} A + B$.

$\begingroup$ Will  You have a 1 in the exponent, when it shouldn't be there. $\endgroup$ – Dustin G. Mixon May 21 '13 at 20:48