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Given a list of complex eigenvalues, where the eigenvalues' complex-conjugates are also in the list, how can I construct a matrix with real entries that has this list as its spectrum?

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    $\begingroup$ It suffices to solve this problem for a single complex eigenvalue and its conjugate. Do you know the eigenvalues of a rotation matrix? $\endgroup$ – Qiaochu Yuan Aug 27 '16 at 21:52
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    $\begingroup$ Note that the block matrix $$ \pmatrix{A&0\\0&B} $$ has the eigenvalues from $A$ combined with the eigenvalues from $B$. $\endgroup$ – Omnomnomnom Aug 27 '16 at 21:57
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    $\begingroup$ @user58865: almost; you need to multiply by $r$ after putting your pairs of complex eigenvalues into polar form. But yes, that's the argument I had in mind. $\endgroup$ – Qiaochu Yuan Aug 27 '16 at 22:41
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  • generate the characteristic polynomial $$p(x):=\Pi_{k=1}^n(x-\lambda_k)(x-\bar \lambda_k)=\Pi_{k=1}^n(x^2-2 (\Re{\lambda_k})x+|\lambda_k|^2)$$ which has real coefficients.

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  • Compute its characteristic polynomial $p$ with the leading coefficient $1$ and other coefficients $a_i$.
  • Than, your matrix is

$$ \left[\begin{matrix} & 1 & & & \\ & & 1 && \\ & & & \ddots& \\ & & & & 1 \\ a_{n - 1} & & \cdots& a_1 & a_0 \end{matrix}\right] $$

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