2008 July « Martin’s Blog

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David Barry is famous on the internet

July 27, 2008 at 6:38 am | In Cricket | 11 Comments

See here. At this page you can even see cute little copyright David Barry signs..

But seriously congratulations Dave.

The kind of things I’m doing at the moment

July 13, 2008 at 7:30 pm | In Mathematics | 4 Comments

Some notes and an example (question 1A on the January ‘08 Algebra Qual).

Given $A \in M_n(F)$ the fundamental theorem of finitely generated modules over a PID gives us the ability to write a rational canonical form which is a block diagonal matrix with blocks $C_{a_1(x)},\ldots,C_{a_m(x)}$ where $C_{a_i(x)}=\begin{bmatrix} 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{n-1} \\ \end{bmatrix}$
is the companion matrix whose characteristic polynomial is $a_i(x) = x^n+c_{n-1}x^{n-1}+\ldots+c_0$ and the $a_i$ are chosen to be monic and satisfy $a_1 \mid \cdots \mid a_m$ .

Then we have the following facts:

The final factor $a_m$ is the minimal polynomial of $A$ .
The product of the $a_i$ is the characteristic polynomial of $A$ .
The matrix is similar to its rational canonical form and this form is unique.
The rcf is the same even if we calculate it over a larger field.

Question 1A.
Let $A$ be an element of finite order in $GL(2,\mathbb{Q})$ . Find all possible orders and give an example illustrating each possible order.

The matrix $A$ has a rational canonical form $C$ such that $A = UCU^{-1}$ . Then $A^n = UC^nU^{-1}$ so $A$ and $C$ have the same order. Thus we can now assume that $A$ is in its rational canonical form.

Now, $A$ can either have one or two blocks. If it has two blocks then it must come from $a_1 \mid a_2$ both of which are monic degree one polynomials so must be equal. Therefore if $a_1 = a_2 = x-c$ we have $A=\begin{bmatrix} c & 0 \\ 0 & c \end{bmatrix}$ so if $A$ is finite order it can be order 1 or 2 where $A = I, -I$ respectively.

Otherwise $A$ is the companion matrix whose characteristic and minimal polynomial is $a(x)$ . If $A^n=I$ then $a(x) \mid x^n-1$ . But $x^n-1 = \prod_{d\mid n} \Phi_d (x)$ so we are looking for degree two cyclotomic polynomials. Now the degree of $\Phi_d$ is $\phi(d)$ . We can check $\phi(d)$ for small numbers (up to 12 say) and see that $\phi(d)=2$ only for $d=3,4,6$ in this range. For $d > 12$ notice that $d$ is divisible either by a prime bigger than or equal to 5, by 8 or by 9. In each of these cases we get a factor bigger than 2 in $\phi(d)$ .

Now we show that the order of the companion matrix, $C$ , of $\Phi_d$ is $d$ . The minimal polynomial of $C$ is $\Phi_d$ which divides $x^d-1$ so $C^d=I$ . Also if $C^n=I$ then $\Phi_d(x) \mid x^n-1$ which means that $d \mid n$ (here we use the fact that $\Phi_d$ is irreducible in $\mathbb{Z}[x]$ , a UFD). Thus $d$ is the order of $C$ .

Finally, calculate that $\Phi_3(x) = x^2+x+1$ , $\Phi_4(x) = x^2+1$ and $\Phi_6(x) = x^2-x+1$ so examples of matrices of order 3, 4 and 6 are $\begin{bmatrix} 0 & -1 \\1 & -1 \\\end{bmatrix}$ , $\begin{bmatrix} 0 & -1 \\1 & 0 \\\end{bmatrix}$ and $\begin{bmatrix} 0 & -1 \\1 & 1 \\\end{bmatrix}$ .