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So I came across the term canonical multiple times by now, and still dont have a very good idea of what it means. So e.g. a matrix $M$ w.r.t. a canonical basis $B$. What is makes a basis canonical? What does the word even mean?

onimoni
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    Canonical has a technical meaning, but here is just means the "natural basis" within the given context. For example, if working in Euclidean space the "canonical basis" would be the "standard basis" ${e_1,\ldots,e_n}$ with $(e_1\lvert\cdots\lvert e_n)=I_n$. – Alex Youcis Sep 11 '13 at 08:04
  • @AlexYoucis Yeah I figured, but what is that technical meaning (and how does it relate to this case)?. There are dozens of other terms in which canonical is used. – onimoni Sep 11 '13 at 08:08
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    @AlexYoucis You mean, "natural" has a technical meaning. :p – Zhen Lin Sep 11 '13 at 08:10
  • @user61001 The actual definition of canonical is a little sophisticated, and it doesn't really apply in this case (although it probably applies in other cases you're thinking of). The word is just misused, in some sense, to mean "natural", "obvious", or "dictated by circumstances". – Alex Youcis Sep 11 '13 at 08:10
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    @ZhenLin Can't we naturally identify "natural" and "canonical"? :) – Alex Youcis Sep 11 '13 at 08:11
  • You might wanna check out this mo question: http://mathoverflow.net/questions/19644/what-is-the-definition-of-canonical - I really like the top answer, even though it's not very mathematical ;) – roman Sep 11 '13 at 09:17

2 Answers2

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Suppose we have a mathematical object. There can be many ways of representing an object that are equivalent to this object for the purposes of solving some problem.

Rather than solve a given problem for all possible objects, we often only need to solve the problem for one representative from each equivalence class. Representatives from these equivalence classes can be called canonical; and it is sufficient to solve the problem only for canonical representatives.

We usually choose canonical representatives that are easy for us to work with.

For example, for graphs, we can sometimes choose a specific way of labeling the vertices. These graphs $$(\{1,2,3\},\{12,13\})$$ $$(\{x,y,z\},\{xy,xz\})$$ and $$(\{3,2,1\},\{32,31\})$$ are all structurally the same graphs, but have different labeled vertices. Canonical labeling the graph gives a specific representative from each isomorphism class of graphs.

We might even allow equivalence classes to have more than one canonical representative. Solving the problem for all canonical representatives nevertheless still amounts to solving the problem for all objects.

As another example, consider Latin squares. The Latin square $$\begin{bmatrix} A & B & C \\ C & A & B \\ B & C & A \\ \end{bmatrix}$$ and $$\begin{bmatrix} A & B & C \\ B & C & A \\ C & A & B \\ \end{bmatrix}$$ formed by swapping the last two rows might be considered equivalent (for a given purpose). We see the second one is in reduced form, i.e., the first row and first column are in the same order. So, we might regard this as a canonical form.

However, in the Latin square case, there are usually many ways to permute the rows and columns (and symbols) to get the first row and first column in order. So there would be many reduced (or canonical) representatives from each equivalence class.

Rebecca J. Stones
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Canonical form is a bit the normal form

$ 3 / 6 = \frac{3}{6} $ but your lecturer would expect you to answer
$ 3 / 6 = \frac{1}{2} $ because $\frac{1}{2} $ is the canonoical form.

in principle if your answers didn't have to be canonical you could answer every question by repeating the exercise.

Willemien
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