Editing Characterization (mathematics)

{{Short description|Term in mathematics}}
In [[mathematics]], a '''characterization''' of an object is a set of conditions that, while possibly different from the definition of the object, is logically equivalent to it.<ref name=":0">{{Cite web|url=http://mathworld.wolfram.com/Characterization.html|title=Characterization|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-21}}</ref> To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X'' have [[property (philosophy)|property]] ''P'', but that ''X'' is the ''only'' thing that has property ''P'' (i.e., ''P'' is a defining property of ''X''). Similarly, a set of properties ''P'' is said to characterize ''X'', when these properties distinguish ''X'' from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of ''X'' in terms of ''P'' include "''P'' is [[necessary and sufficient]] for ''X''", and "''X'' holds [[if and only if]] ''P''".

It is also common to find statements such as "Property ''Q'' characterizes ''Y'' [[up to]] [[isomorphism]]". The first type of statement says in different words that the [[extension (semantics)|extension]] of ''P'' is a [[singleton (mathematics)|singleton]] set, while the second says that the extension of ''Q'' is a single [[equivalence class]] (for isomorphism, in the given example &mdash; depending on how ''[[up to]]'' is being used, some other [[equivalence relation]] might be involved).

A reference on mathematical terminology notes that ''characteristic'' originates from the Greek term ''kharax'', "a pointed stake":<blockquote>From Greek ''kharax'' came ''kharakhter'', an instrument used to mark or engrave an object. Once an object was marked, it became distinctive, so the character of something came to mean its distinctive nature. The Late Greek suffix ''-istikos'' converted the noun ''character'' into the adjective ''characteristic'', which, in addition to maintaining its adjectival meaning, later became a noun as well.<ref>Steven Schwartzmann (1994) ''The Words of Mathematics: An etymological dictionary of mathematical terms used in English'', page 43, [[The Mathematical Association of America]] {{ISBN|0-88385-511-9}}</ref></blockquote>Just as in chemistry, the [[characteristic property]] of a material will serve to identify a sample, or in the study of materials, structures and properties will determine [[characterization (materials science)|characterization]], in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in ''[[Mathematical Reviews]]'', as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review.<!-- Might consider a different reference, since the access to Mathematics Reviews requires active subscription . -->

In an arbitrary context of objects and features, characterizations have been expressed via the [[heterogeneous relation]] ''aRb'', meaning that object ''a'' has feature ''b''. For example, ''b'' may mean [[abstract and concrete|abstract or concrete]]. The objects can be considered the [[extension (semantics)|extension]]s of the world, while the features are expressions of the [[intension]]s. A continuing program of characterization of various objects leads to their [[categorization]].

== Characterizations in higher mathematics ==
Characterizations are particularly important in higher mathematics, where they take up a large volume of theory in typical undergraduate courses. They are commonly known as "necessary and sufficient conditions," or "if-and-only-if statements." Characterizations help put difficult objects into a form where they are easier to study, and many types of objects in mathematics have multiple characterizations. Sometimes, one characterization in particular particular is more readily generalizable to abstract settings than the others, and it is often chosen as a ''definition'' for the generalized concept.

In [[real analysis]], for example, the [[Completeness of the real numbers|completeness]] property of the real numbers has several useful characterisations:<ref>Abbott, Stephen. ''Understanding Analysis''. New York, Springer, 2016.</ref>

* The [[least-upper-bound property]]
* The [[greatest-lower-bound property]]
* The [[Nested intervals|nested interval property]]
* The [[Bolzano–Weierstrass theorem|Bolzano-Weierstrass theorem]]
* The convergence of [[Cauchy sequence|Cauchy sequences]]

A typical real analysis university course would begin with the first of these, the [[least-upper-bound property]], as an axiomatic definition of the reals (sometimes called the "axiom of completeness" in texts), and gradually prove its way to the last, the convergence of Cauchy sequences. The proofs are quite nontrivial. Among these five characterizations, the Cauchy-sequence perspective turns out to be the easiest to generalize, and is chosen as the ''definition'' for the completeness of an abstract [[metric space]]. However, the least-upper-bound property is often the most useful to prove facts about real numbers themselves, such as the [[intermediate value theorem]]. Thus the most useful and most generalizable characterizations are at times different.

Another example of this phenomenon is found in [[physics]]. [[Hamiltonian mechanics]] is a characterization of classical mechanics, being equivalent to Newton’s laws. However, it is much easier to generalize to [[quantum mechanics]] and [[statistical mechanics]], which is its primary virtue. However, it is a different characterization, [[Lagrangian mechanics]], that is often preferred for the study of classical mechanics itself.<ref>John Robert Taylor. ''Classical Mechanics''. Sausalito, Calif., University Science Books, Cop, 2005.</ref>

Since a characterization result is equivalent to the initial definition or axiom(s) of the object, it can be used as an equivalent definition, from which the original definition can be proved as a theorem. This leads to the question of which definition is “best” in a given situation, out of many possible options. There is no absolute answer, but the ones that are chosen by authors of books or papers is often a matter of aesthetic or pedagogical considerations, as well as convention, history, and tradition. For real numbers, the least-upper-bound property may have been chosen on the grounds of being easier to learn than Cauchy sequences.

One of the most important results in [[complex analysis]] is a characterization result, namely the fact that all locally complex-differentiable functions are analytic (equal to their Taylor series).<ref>Brown, James, and Ruel Churchill. ''Complex Variables and Applications''. McGraw-Hill Science/Engineering/Math, 2009.</ref>

Characterisations are very common in [[abstract algebra]], where they often take the form of “structure theorems,” expressing the structure of an object in a simple form. These results are often very difficult to prove. In the theory of matrices, the Jordan Canonical Form is a characterization, or structure theorem, for complex matrices,<ref>Axler, Sheldon. ''Linear Algebra Done Right''. Springer Nature, 28 Oct. 2023.</ref> and the spectral theorem is likewise for symmetric matrices (if real) or Hermitian matrices (if complex). According to the spectral theorem, the real symmetric matrices are precisely the ones that have a basis of perpendicular eigenvectors<ref>Axler, Sheldon. ''Linear Algebra Done Right''. Springer Nature, 28 Oct. 2023.</ref> (called principal axes in physics). In the theory of [[Group (mathematics)|groups]], there is a structure theorem for [[Finite abelian group|finite abelian groups]], that states that every such group is a [[Direct product of groups|direct product]] of [[Cyclic group|cyclic groups]].<ref>David Steven Dummit, and Richard M Foote. ''Abstract Algebra''. Danvers, John Wiley & Sons, 2004.</ref>

As if-and-only-if statements, characterizations are, in a sense, the “strongest” type of mathematical theorem, which is in line with the difficulty of their proofs. Consider a generic mathematical theorem, that A implies B. If B does not imply A, the theorem may be said to be “underpowered”, as the proved statement B is “weaker” than the ingredient A, being not strong enough to prove A on its own. In a characterization, however, B must imply A also – the proved statement is as strong as the ingredient, and it can be no stronger. In a sense, such a result “uses” all of the structure in A in proving B.

==Examples==
* A [[rational number]], generally defined as a [[ratio]] of two integers, can be characterized as a number with finite or repeating [[decimal expansion]].<ref name=":0" />
*A [[parallelogram]] is a [[quadrilateral]] whose opposing sides are parallel. One of its characterizations is that its diagonals bisect each other. This means that the diagonals in all parallelograms bisect each other, and conversely, that any quadrilateral whose diagonals bisect each other must be a parallelogram.
* "Among [[probability distribution]]s on the interval from 0 to ∞ on the real line, [[memorylessness]] characterizes the [[exponential distribution]]s." This statement means that the exponential distributions are the only probability distributions that are memoryless, provided that the distribution is continuous as defined above (see [[Characterization of probability distributions]] for more).
* "According to [[Bohr–Mollerup theorem]], among all functions ''f'' such that ''f''(1) = 1 and ''x f''(''x'') = ''f''(''x'' + 1) for ''x'' > 0, log-convexity characterizes the [[gamma function]]."  This means that among all such functions, the gamma function is the ''only'' one that is [[log-convex]].<ref>A function ''f'' is ''log-convex'' [[Iff|if and only if]] log(''f'') is a [[convex function]]. The base of the logarithm does not matter as long as it is more than 1, but mathematicians generally take "log" with no subscript to mean the [[natural logarithm]], whose base is ''e''.</ref>
* The circle is characterized as a [[manifold]] by being one-dimensional, [[compact space|compact]] and [[connected space|connected]]; here the characterization, as a smooth manifold, is [[up to]] [[diffeomorphism]].

== See also ==
* {{annotated link|Characterizations of the category of topological spaces}}
* {{annotated link|Characterizations of the exponential function}}
* {{annotated link|Characteristic (algebra)}}
* {{annotated link|Characteristic (exponent notation)}}
* {{annotated link|Classification theorem}}
* {{annotated link|Euler characteristic}}
* {{annotated link|Character (mathematics)}}

==References==
{{Reflist}}

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[[Category:Mathematical terminology]]
[[Category:Equivalence (mathematics)]]