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Let $A$, $B$ be two $n\times n$ matrices over $\mathbb{C}$. If $A$, $B$ have the same minimal polynomials and characteristic polynomials, then can we prove that $A$, $B$ are conjugate? i.e., does there exist $P\in GL(n;\mathbb{C})$ such that $PAP^{-1}=B$?

Prove or give counterexamples.

Shiquan
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  • Unless you're limiting yourself to invertible matrices and working with that group, for general matrices the term is similar matrices, not conjugate ones. – DonAntonio Dec 04 '13 at 12:55
  • Indeed so, @DustanLevenstein. That "less than" should have been "less", just as I myself wrote here http://math.stackexchange.com/questions/198857/are-two-matrices-having-the-same-characteristic-and-minimal-polynomial-always-si – DonAntonio Dec 04 '13 at 20:30

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Nope. Consider two $4 \times 4$ matrices:

$$\begin{pmatrix}0 & & & \\ & 0 & & \\ & & 0 & 1\\&&0&0\end{pmatrix}$$

$$\begin{pmatrix}0 & 1 & & \\ 0 & 0 & & \\ & & 0 & 1\\&&0&0\end{pmatrix}$$

(where I've written the matrices in Jordan block form; the omitted entries are zero.) Both matrices have characteristic polynomial $x^4$ and minimal polynomial $x^2$, but they are not similar.

Dustan Levenstein
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