Given complex eigenvalues, with its complex-conjugates also as eigenvalues, how can I construct a matrix with real entries?

Question

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?

Answer 1

• 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.

  • then take the companion matrix associated with $p(x)$ (https://en.wikipedia.org/wiki/Companion_matrix).

Answer 2

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

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