Is there a matrix with real entries such that its minimal polynomial with coefficients in the complex field is different than its minimal polynomial over the real numbers?
Minimal polynomial over the complex field
1 Answers
No, the minimal polynomial is independent of the field over which it is calculated, given that the entries are in the field. The minimal polynomial can, of course, be reducible over $\mathbb{C}$, while it is irriducible over $\mathbb{R}$.
To prove this fact, let $m_A^\mathbb{C}(x)$ denote the minimal polynomial of $A$ over $\mathbb{C}$. By the definition of $m_A^\mathbb{C}(x)$, it is the polynomial with the lowest degree such that $m_A^\mathbb{C}(A)=0$ and the leading coefficient is 1. Now, if there is a complex coefficient in $m_A^\mathbb{C}(x)$, then look at $m_A^\mathbb{C}(x)-\overline{m_A^\mathbb{C}(x)}$ (complex conjugate). This will be a non-zero polynomial of smaller degree which will be zero at $A$. That is a contradiction.
Hence all the coefficients of $m_A^\mathbb{C}(x)$ are real, so $m_A^\mathbb{C}(x)=m_A^\mathbb{R}(x)$
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2Perhaps you should also remind the reader that as $A$ has real entries, it will also be a zero of $\overline{m_{A}^{\mathbb{C}}(x)}$. – 2012-06-14