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Let $A\in M_{n}(\mathbb{C})$ and let $\{\lambda_{1},...,\lambda_{n}\}$ be the eigenvalues of $A$. Is it true that the eigenvalues $\{\mu_{1},...,\mu_{n}\}$ of $\frac{A+\bar{A}}{2}$ are of the form $\mu_{i}=\frac{\lambda_{p}+\bar{\lambda_{q}}}{2}$, for some $p,q\in\{1,...,n\}$?

I have the same question on the imaginary part of $A$.

If it is true, then spectral mapping theorem might be useful for its proof.

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Consider the matrix $A = \left[\begin{matrix}1 & i\\ -i & 1\end{matrix}\right]$. It has eigenvalues $0$ and $2$, but the real part of $A$ is the identity matrix which does not have $2$ as an eigenvalue.

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By unitary, I mean $AA^{\ast}=A\bar{A}^{t}=I$; $A$ is not a unitary. –  user78800 Jun 9 '13 at 3:32
    
Sorry, I was thinking of hermitian ($\bar{A}^t = A$). –  Michael Albanese Jun 9 '13 at 3:35
    
Even in your counterexample, we have $\frac{2+0}{2}=1$. So, lets rephrase the question. Is it true that $Spec((A+\bar{A})/2)=(Spec(A)+Spec(\bar{A}))/2$? –  user78800 Jun 9 '13 at 3:41
    
I don't see how $\frac{2+0}{0}$ is relevant. Anyway, if you want to change the question, I recommend editing your original post. –  Michael Albanese Jun 9 '13 at 3:53
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