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Suppose $A$ is a real $n\times n$ normal matrix, and consider the matrix $$B=(I_n-\theta A)^{-1}(A+A^T)(I_n-\theta A^T)^{-1},$$ where $\theta$ is a real scalar such that $(I_n-\theta A)$ is invertible.

Since $B$ is symmetric, I can write $$B=H\Lambda H^T,$$ where $\Lambda$ is a diagonal matrix containing the eigenvalues of $B$ and $H$ is an orthogonal matrix containing real eigenvectors of $B$.

My question is: how is $H$ related to the eigenvectors of $A$, or of ($A+A^T$)?

(Letting $A=UDU^{*}$, where $U$ is unitary and $D$ is the diagonal matrix containing the eigenvalues $d_i$ of $A$, we see that the eigenvalues of $B$ are $|1-\theta d_i|^{-1}\mathrm{Re}(d_i)$. It's the eigenvectors I'm not sure about. )

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There's something wrong with your question. Let $A=I_n$, $A$ is a normal matrix, but $I_n-A$ isn't invertible. –  Git Gud Feb 6 '13 at 18:40
    
ok thanks, edited –  mark Feb 6 '13 at 18:54
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