Real matrices have real minimal polynomials ($\overline{m_A}=m_A(A\in M_n(\mathbb{R}))$)

The minimal polynomial of a real matrix, initially computed over the complex numbers, has real coefficients. Its coefficientwise complex conjugate is also a monic polynomial of the same degree annihilating the matrix. Uniqueness of the minimal polynomial forces the two polynomials to agree.