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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 !

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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 $uA+B=(u-1)A+(A+B)$, and so the triangle inequality gives $$ |u-1|\|A\|_2-1 \leq\|uA+B\|_2 \leq|u-1|\|A\|_2+1. $$ That is, the largest eigenvalue of $uA+B$ is on the order of the (unknown) largest eigenvalue of $A$.

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Let $C=A+B$ be any real matrix. Let $A$ be any real matrix, then $B=C-A$ 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 $(1-e^{i\theta})$ times a real number. Since the trace of $A+B$ is a real number between $d$ and $2-d$, with $d$ the dimension of the matrices, this gives you a band of possible values for the trace of $e^{i\theta} A + B$.

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

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