Algorithm for real matrix given the complex eigenvalues

Question

Given complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a real matrix which has these eigenvalues. I know the matrix is not unique as eigenvectors are not fixed but in my case any real matrix will suffice.

Answer 1

If eigenvalue is $a \pm bi$, the matrix is $$ \begin{bmatrix} a & -b \\ b & a \end{bmatrix}. $$

For more eigenvalue pairs, place $2 \times 2$ blocks like this one down the diagonal of your $2k \times 2k$ matrix.

Answer 2

If the given eigenvalues are all distinct, then you can take the companion matrix of the polynomial $(x-\lambda_1)\cdots(x-\lambda_n)$, which has real coefficients because the roots occur in conjugate pairs.