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How do I represent algebraic numbers as matrices?

pdmclean
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  • All algebraic numbers? As matrices of what order? – lhf Sep 21 '15 at 12:47
  • Which properties do you want this representation to preserve? For example, if $A_x$ is the matrix representing the number $x$, then is $A_{x+y}$ supposed to be $A_x+A_y$? Same for multiplication? – Klaus Draeger Sep 21 '15 at 12:53
  • Matrices have a minimal polynomial, algebraic numbers have a defining polynomial. I would think that's where you're supposed to connect them. – Arthur Sep 21 '15 at 12:53

1 Answers1

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Let $L/K$ be a field extension of finite degree $n$.

Take $a \in L$ and consider the map $\mu_a: x \mapsto ax$ on $L$. This is a $K$-linear transformation when you consider $L$ as a vector space over $K$.

Fix a basis $B$ of $L/K$ and express $\mu_a$ as a matrix with respect to this basis.

Then $\phi: L \to M_n(K)$ given by $\phi(a)=[\mu_a]_B$ is a ring homomorphism and so is injective. This gives a faithful representation of $L$ as a ring of matrices over $K$.

When $L=K(a)$ and $B=\{1,a,\dots,a^{n-1}\}$, the matrix you get is the companion matrix of the minimal polynomial of $a$.

lhf
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  • See https://math.stackexchange.com/questions/1444284/how-abi-becomes-left-matrixa-b-b-a-right for recent related question. – lhf Sep 21 '15 at 12:56
  • You should note that $L$ is essentially being treated as a vector space. You should say the word "vector space" at least once. :) – Thomas Andrews Sep 21 '15 at 12:57
  • @ThomasAndrews, done, thanks for the nudge. – lhf Sep 21 '15 at 12:58
  • See also http://math.stackexchange.com/questions/1441614/are-there-any-fields-with-a-matrix-representation-other-than-mathbbc. – lhf Sep 22 '15 at 02:37