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Let $A$ be an $n \times n$ matrix over an infinite field $\mathbb{F}$ (e.g., $\mathbb{R}$ or $\mathbb{C}$). Suppose $A$ is not a scalar matrix (i.e., $A \neq cI$ for any $c \in \mathbb{F}$). Prove that the similarity class of $A$, defined as $$ [A] := \left\{ P^{-1} A P \mid P \in \operatorname{GL}_n ({\Bbb F}) \right\},$$ is infinite.

I would like a proof (or reference) that leverages group theory, particularly the action of $\operatorname{GL}_n(\mathbb{F})$ by conjugation on $M_n(\mathbb{F})$. Here are some thoughts:

  1. The stabilizer of $A$ under conjugation is $C(A) = \{ P \in \operatorname{GL}_n(\mathbb{F}) \mid P^{-1}AP = A \}$, i.e., the centralizer of $A$.

  2. If $A$ is not scalar, $C(A)$ is a proper subgroup of $\operatorname{GL}_n(\mathbb{F})$. Does this imply that the orbit $[A]$ is infinite?

  3. Over infinite fields, can we exploit the infinite index of $C(A)$ in $\operatorname{GL}_n(\mathbb{F})$ to conclude that $[A]$ is infinite?

P.S. I'm new here, so please point out any issues with the formatting or the question itself.

Rodrigo de Azevedo
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MathPacer
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jun 09 '25 at 17:52
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    Do you know Burnside’s lemma? That combined with what you know is enough. – Malady Jun 09 '25 at 18:01

1 Answers1

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If $A$ is not scalar, then there exists $v \in F^n$ such that $w = Av$ is not a scalar multiple of $v$, so $v$ and $w$ are linearly independent.

For any scalar $\lambda \in F$, we can choose $P_\lambda \in {\rm GL}(n,F)$ such that $P_\lambda v=v$ and $P_\lambda w = w+\lambda v$.

Then $P_\lambda AP_\lambda ^{-1}v = w + \lambda v$, so distinct $\lambda \in F$ give distinct matrices $P_\lambda AP_\lambda^{-1}$.

Derek Holt
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