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The sense of norm, standard, or archetype has been used in many disciplines. By contrast, the existence of Jordan canonical form for a matrix is a deep theorem.Īccording to OED and LSJ, the term canonical stems from the Ancient Greek word kanonikós ( κανονικός, "regular, according to rule") from kanṓn ( κᾰνών, "rod, rule").
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For example, polynomials are conventionally written with the terms in descending powers: it is more usual to write x 2 + x + 30 than x + 30 + x 2, although the two forms define the same polynomial. The uniqueness requirement is sometimes relaxed, allowing the forms to be unique up to some finer equivalence relation, such as allowing for reordering of terms (if there is no natural ordering on terms).Ī canonical form may simply be a convention, or a deep theorem. Operations on classes are carried out by combining these representatives, and then reducing the result to its least non-negative residue. For example, in modular arithmetic, the canonical form for a residue class is usually taken as the least non-negative integer in it.
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There is also a practical, algorithmic question to consider: how to pass from a given object s in S to its canonical form s*? Canonical forms are generally used to make operating with equivalence classes more effective. In practical terms, it is often advantageous to be able to recognize the canonical forms. Property 3 is redundant it follows by applying 2 to 1.
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In this context, a canonical form is a representation such that every object has a unique representation (with canonicalization being the process through which a representation is put into its canonical form).
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In computer science, and more specifically in computer algebra, when representing mathematical objects in a computer, there are usually many different ways to represent the same object.
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