What is the difference between a definition and a characterization?

Question

In an exam I was asked to define the unitary group $U_n(\mathbb{C})$

where I have answered $$U_n(\mathbb{C})=\{A \in \mathbb{C}^{n,n} | AA^H=I\}$$

However I did not get full point count because what I have delivered was a characterization and not a definition. The definition in my script is given as:

Let $(V, s)$ be unitary, $V$ finitely dimensional. Then $$U(V,s):=GL(V,s)$$ with $GL(V,s)$ being the set of all metric endomorphisms on $V$.

Define $$U_n(\mathbb{C}) := U(\mathbb{C} ^n , s^n ).$$

It is true that

$$U_n(\mathbb{C}) = \{A \in GL_n(\mathbb{C}) | A^{ −1} = A^H \}$$

I am quite confused and not sure if I should argue with my professor. So what is the difference between what I have given as the definition, which supposedly is only a characterization and what exactly is the difference in general?

Answer 1

Edit: I wrote a longer and more opinion based answer, but I think this shorter argument is better and sufficient to justify your professor.

I think that from a conceptual point of view, it is not completely correct to say that an element in $GL(V)$ is a matrix. It is true that it can be represented by one, and it is true that you can characterize morphisms by conditions on the matrices that do not depend on the choice of a basis. But I still think it is not conceptually correct to say that a linear map is a matrices.

Answer 2

You should ask for clarification; your professor is using the terms differently from what I believed was a perfectly standard (if obscure) way, as follows.

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A definition is a way of crisply pinning down some object or concept. A characterisation of something is a definition of that thing by noting some property that it uniquely satisfies.

For example, the group $\mathbb{Z}_5$ can be defined up to isomorphism as "the group $\{0,1,2,3,4\}$ with addition performed mod $5$", or it may be characterised (and hence defined) up to isomorphism as "the unique group with five elements". The former is a definition but not a characterisation: it is an explicit construction of the desired object. The latter is a characterisation and a definition: it identifies the object uniquely by means of identifying some property of it (in this instance, its cardinality).

Answer 3

I think it is somewhat pointless to insist too heavily on any attempt to distinguish "definition" from "characterization", since in colloquial English they are indistinguishable.

In mathematical English, post-Bourbaki and post-set-theory, "definition" came to have a "charged" sense of "definition in terms of sets (and/or set theory)", as opposed to any other sort of characterization. In particular, as it developed, these definitions-as-sets were/are often not really to the point, e.g., the (nevertheless interesting) "definition" of an ordered pair $(a,b)$ as the set $\{\{a\},\{a,b\}\}$, and such.

The much-later-developing categorical style of "definition" of classical objects lost much of its set-theoretic-definitional style, and instead characterized objects "indirectly" in terms of their interactions with other objections in the category-theory world. (At the same time, the formal version of category theory had "definitions" of its own, if not always set-theoretic.)

The very-informal distinction I attempt to make (especially in graduate courses) is between an uninformative austere "definition", and an informative (usually more category-theoretic) non-set-theoretic "what does it do?" characterization. Indeed, many of the traditional definitions only seem interesting when one has already seen many examples. E.g., "group" is not interesting so much because it's a simple case of set with binary operation satisfying a few axioms, but because this captures some essential features of many things we would have already seen.

Yes, there still is a tradition that insists that if we've not been given "a precise definition", which somehow implicitly requires a set-theoretic construction, we've been short-changed, or the thing hasn't been described. I think this is misleading (e.g., is a real number a Cauchy sequence of rationals mod null, or is it a Dedekind cut? Maybe neither?).

Genuine "characterization" is better, in general, because it does confess intent, rather than pretending some sort of inert dispassion.

EDIT: and I forgot to mention the very important feature of a good "characterization" (in a category-theoretic sense) is that it usually proves that there is a unique such thing (up to unique isomorphism). Thus, varying (set-theoretic) constructions inevitably produce the same thing, up to isomorphism anyway. In particular, there's no room for whim. For example, the categorical definition of "product" of topological spaces proves uniqueness, so that there is no room for discussion of "defining" a finer topology than the usual "definition", despite one's possible initial reaction that it is disappointingly coarse in the case of infinitely-many factors. And, saaaay, why is it the cartesian product of the underlying sets, anyway? :) And so on.