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"Canonical" and "natural" are two words frequently seen in mathematical literature. For example, we often find "there is no canonical/natural way to", "it's canonical/natural to". So I'd like to know what is exactly "canonical/natural" way and also some examples for explain.

zhangwfjh
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    Usually they're just colloquial and have no specific definition. Occasionally 'natural' is used in the category theoretic sense (see here). – Clive Newstead Oct 08 '14 at 04:47
  • @CliveNewstead Thanks, suppose "natural way to identify two things" means there's natural transformation between them, is there any quick method to know why? Or how to find the transformation or disprove the existence? – zhangwfjh Oct 08 '14 at 04:55

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"Canonical" can mean simple in appearance or utility. For example, The Jordan Canonical Form is a transformation of a matrix so that it's block diagonal and all the blocks are upper triangular. It's simple in appearance (most of the elements are 0, that's good). And it's a nice form to be able to use. For example, it's simple to find a solution to $J x = b$ if J is the Jordan Canonical Form of a matrix.

NicNic8
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