Let $A$ be a $n\times n$ matrix over $\mathbb{C}$ satisfying $X^m-1$ for some positive integer $m$.
$a)$ Show that $A$ is diagonalizable over $\mathbb{C}$.
$b)$ Show that $tr(A)$ is a sum of roots of $X^m-1$.
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Can you provide any thoughts, work, motivation, or context? – Michael Burr May 07 '17 at 15:54
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Hints: (1) All eigenvalues of $A$ are $m$th roots of unity. (2) If you know the Jordan form, can you prove that all blocks are size $1$ by contradiction? – Michael Burr May 07 '17 at 15:57
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What have you tried? Were are you getting stuck? Do you know what an eigenvalue is? Do you know what a minimal polynomial is? – Ben Grossmann May 07 '17 at 15:57
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A basic theorem of linear algebra is that a matrix is diagonalizable iff its minimal polynomial is the product of distinct linear factors.
Since the ground field is $\mathbb{C}$ the polynomial $x^n-1$ factors into linear factors and its well know that they are distinct ( because the polynomial is prime to its derivative $nx^{n-1}$). Thus $A$ is diagonalizable as its minimal polynomial must divide this polynomial.
Rene Schipperus
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