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Suppose we have a finite group $G$ and and an $n$-dimensional vector space $V\cong \Bbb C^n$ over the field $\Bbb C$ of complex number. My professor said the other day that for every group element $g$ in $G$ and for every representation $\rho:G\to\text{GL}_n(\Bbb C)$ of $G$ on $V$, the matrix $\rho(g)\in\text{GL}_n(\Bbb C)$ is diagonalizable. I'm having trouble convincing myself that this is true. The text below is from a website that agrees with what my professor said:

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I believe that all the eigenvalues of $M$ must be $n$-th roots of unity, and that each eigenvalue appears only once as root of $M$'s minimal polynomial. I am having trouble convincing myself, however, that this means $M$ is diagonalizable. How do we know that the eigenvectors of $M$ span the vector space $V$? Couldn't $M$ have a minimal polynomial of degree less than $n$? Any help would be much appreciated.

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  • $\begingroup$ I thought that representation matrices are in block-triangular form, not exactly in triangular form, but we do not need that remark for the argument, as the theorem from linear algebra goes: a matrix is diagonalizable iff its minimal polynomial has no multiple roots $\endgroup$
    – AgentSmith
    Sep 18, 2018 at 18:46

2 Answers 2

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The theorem that solves this is (indeed, from linear algebra):

Theorem: a matrix over some field $\;F\;$ is diagonalizable iff its minimal polynomial is the product of different linear factors.

Now, we know that over an algebraic closed field any polynomial decomposes in a product of linear factors, and thus this product will be of different linear factors iff the poloynomial has no repeated roots (i.e., if all its roots are simple).

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    $\begingroup$ Thanks! Now I just have to figure out why that theorem is true... $\endgroup$
    – Jasha
    Oct 12, 2014 at 14:46
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Adding a comment to the answer of Timbuc to answer Jasha comment (which is already exhaustive). All you have to know is that the minimal polynomial of the matrix, let's called it $\mu_M(t) \mid \mu_M(t)$ (standard argument using $\mu_M(t) \mid p_M(t)$, where $p$ denotes the characteristical polynomial), and the thesis know is a consequence of Timbuc answer, since if we divide a polynomial with all distinct roots, we must be a finite product of some of them, and so our roots are distinct.

Edit: Actually we don't need the "standard argument", it is sufficient to know that $x^{n}-1$ is in the ideal generated by $\mu_M(t)$.

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