We show that given a polynomial, one can (without knowing the roots) construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed. For a real polynomial, the matrix can have purely imaginary off-diagonal entries. For a polynomial with only real distinct roots, the matrix can be chosen real. © 1990.
Fiedler, M. (1990). Expressing a polynomial as the characteristics polynomial of a symmetric matrix. Linear Algebra and Its Applications, 141(C), 265–270. https://doi.org/10.1016/0024-3795(90)90323-5