What is meant by canonical?

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?

Solution 1:

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.

Solution 2:

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.