Classify up to similarity all 3 x 3 complex matrices $A$ such that $A^n$ = $I$.
classification up to similarity of complex n-by-n matrices
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0Alex, you're right. I've been hesitant to post answers like this because it feels like I'm whoring for points. But I guess comments can get you points too... – 2010-12-27
3 Answers
If you define $p=X^n-1\in\mathbb C[X]$, then $p(A)=0$. This tells you that the minimal polinomial $m_A$ of $A$ divides $p$ and, in particular, that $m_A$ has all its roots simple, because the same is true of $p$.
It follows that $A$ is diagonalizable, so, up to similarity, you can suppose that it is diagonal.
Can you see which are the diagonal matrices $A$ which satisfy the condition $A^n=I$?
NB: This argument does not depend on your knowing about Jordan canonical forms.
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5@Josh: Which are the $1\times 1$ **diagonal** matrices such that $A^5=I$? Make a list. When you are done: Which are the $2\times 2$ **diagonal** matrices such that $A^5=I$? Make a list ... If you keep going, I assure you that before you get to $10\times 10$ matrices you'll know the general answer. I, for one, will not tell you the answer to this :) – 2010-12-09
In fact, one does not need to know the characteristic polynomial in this case. Let the minimal polynomial be $p$, then $p\mid (x^3-1)$. It is important to see that $x^3-1$ has three distinct roots in $\mathbb{C}$. Hence $p$ cannot have repeated roots in $\mathbb{C}$. Thus $A$ must be diagonalizable over $\mathbb{C}$, with each diagonal entry a root of $x^3-1$. Hence $A$ is similar to \begin{equation} \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}, \qquad a,b,c \in \{1,e^{i2\pi/3},e^{-i2\pi/3}\} \end{equation}
It is an Hoffman Kunze exercise problem. It will be for $3\times 3$ matrices $A$, $A^3=I$.
My answer is, the minimal polynomial of $A$ will divide $X^3-1=0$. Now $x^3-1=(x-1)(x-\omega)(x-\omega^2)$ where $\omega^3=1$. So the minimal polynomial can be of the forms
- $m=x-a$
- $m=(x-a)(x-b)$
- $m=(x-a)(x-b)(x-c)$
Now if $m=(x-a)$,then characteristic polynomial of $A$ will be equal to $(x-a)^3$. Hence $A$ is similar to a diagonal matrix with all entries equal.
If $m=(x-a)(x-b)$, then characteristic polynomial of $A$ will be equal to either $(x-a)^2(x-b)$ or $(x-a)(x-b)^2$. Hence $A$ is similar to a diagonal matrix with two entries equal.
If $m=(x-a)(x-b)(x-c)$, then characteristic polynomial of $A$ will be equal to either $(x-a)(x-b)(x-c)$. Hence $A$ is similar to a diagonal matrix with all entries are unequal.
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0You can format your formulas to a more readable form using dollars as in LaTeX. E.g. `$3^{1+2}$` becomes $3^{1+2}$. See this guide for details: http://meta.math.stackexchange.com/q/5020/166535 – 2015-04-20