Diagonal
349251
225542709
2008-07-14T06:14:36Z
Richard Pinch
1902818
/* Matrices */ definition of off-diagonal
{{Mergefrom|Face diagonal|date=September 2007}}
{{Mergefrom|Space diagonal|date=September 2007}}
{{for|the avenue in [[Barcelona]]|Avinguda Diagonal}}
[[Image:Diagonal uhlopricka.jpg|right|220px]]
A '''diagonal''' can refer to a line joining two nonconsecutive vertices of a [[polygon]] or [[polyhedron]], or in contexts any upward or downward sloping line. The word "diagonal" derives from the [[Greek language|Greek]] ''διαγώνιος'' (''diagonios'')<ref>[http://www.etymonline.com/index.php?search=diagonal&searchmode=none Online Etymology Dictionary]</ref>, used by both Strabo<ref>Strabo, Geography 2.1.36-37</ref> and Euclid<ref>Euclid, Elements book 11, proposition 28</ref> to refer to a line connecting two vertices of a [[rhombus]] or [[cuboid]],<ref>Euclid, Elements book 11, proposition 38</ref> and is formed from dia- ("through", "across") and gonia ("angle", related to gony "knee."), later adopted into Latin as diagonus ("slanting line").
In [[mathematics]], in addition to its geometric meaning, a diagonal is also used in [[matrix (math)|matrices]] to refer to a set of entries along a diagonal line.
==Non mathematical uses==
In [[engineering]], a ''diagonal brace'' is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations diagonal braces are often not connected to the corners of the rectangle.
[[Diagonal pliers]] are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or "on a diagonal", hence the name.
A [[diagonal lashing]] is a type of lashing used to bind spars or poles together applied so that the lashings cross over the poles at an angle.
In [[association football]], the [[diagonal (football)|diagonal]] system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch.
== Polygons ==
As applied to a [[polygon]], a '''diagonal''' is a [[line segment]] joining any two non-consecutive vertices. Therefore, a [[quadrilateral]] has two diagonals, joining opposite pairs of vertices. For any [[convex polygon]], all the diagonals are inside the polygon, but for [[re-entrant polygon]]s, some diagonals are outside of the polygon.
Any ''n''-sided polygon (''n'' ≥ 3), [[Convex polygon|convex]] or [[Concave polygon|concave]], has
:<math>\frac{n(n-3)}{2}\, </math>
diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or ''n'' − 3 diagonals.
== Matrices ==
In the case of a [[square matrix]], the ''main'' or ''principal diagonal'' is the diagonal line of entries running from the top-left to bottom-right corners. For example, the [[identity matrix]] can be defined as having entries of 1 on the main diagonal, and 0s elsewhere. The top-right to bottom-left diagonal is sometimes described as the ''minor'' diagonal or ''antidiagonal''. A ''superdiagonal'' entry is one that is directly above and to the right of the main diagonal. If otherwise unqualified, it refers to the one adjacent to the main diagonal. Likewise, a ''subdiagonal'' entry is one that is directly below and to the left of the main diagonal. The ''off-diagonal'' entries are those not on the main diagonal. A ''[[diagonal matrix]]'' is one whose off-diagonal entries are all zero.
==Geometry==
By analogy, the [[subset]] of the [[Cartesian product]] ''X''×''X'' of any set ''X'' with itself, consisting of all pairs (x,x), is called the diagonal, and is the graph of the identity relation. This plays an important part in geometry; for example, the [[fixed point (mathematics)|fixed point]]s of a [[function (mathematics)|mapping]] ''F'' from ''X'' to itself may be obtained by intersecting the graph of ''F'' with the diagonal.
In geometric studies, the idea of intersecting the diagonal ''with itself'' is common, not directly, but by perturbing it within an [[equivalence class]]. This is related at a deep level with the [[Euler characteristic]] and the zeros of [[vector field]]s. For example, the [[circle]] ''S''<sup>1</sup> has [[Betti number]]s 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-[[torus]] ''S''<sup>1</sup>xS<sup>1</sup> and observe that it can move ''off itself'' by the small motion (θ, θ) to (θ, θ + ε). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the [[Lefschetz fixed point theorem]]; the self-intersection of the diagonal is the special case of the identity function.
==See also==
* [[Jordan normal form]]
* [[Main diagonal]]
* [[Diagonal functor]]
* [[Face diagonal]]
* [[Space diagonal]]
==External links==
{{Wiktionarypar|diagonal}}
*[http://www.mathopenref.com/polygondiagonal.html Diagonals of a polygon] with interactive animation
*[http://mathworld.wolfram.com/PolygonDiagonal.html Polygon diagonal] from [[MathWorld]].
*[http://mathworld.wolfram.com/Diagonal.html Diagonal] of a matrix from [[MathWorld]].
*[http://www.diagonalmedia.com Diagonal media] from [[Diagonal]].
==References==
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[[Category:Elementary mathematics]]
[[Category:Greek loanwords]]
[[ar:قطر (رياضيات)]]
[[ast:Diagonal]]
[[cs:Úhlopříčka]]
[[da:Diagonal]]
[[de:Diagonale (Geometrie)]]
[[es:Diagonal]]
[[eo:Diagonalo]]
[[fr:Diagonale]]
[[ko:대각선]]
[[it:Diagonale]]
[[he:אלכסון]]
[[sw:Ulalo]]
[[hu:Átló]]
[[nl:Diagonaal]]
[[ja:対角線]]
[[pl:Przekątna]]
[[pt:Diagonais de um polígono]]
[[qu:Ch'iqan]]
[[ru:Диагональ]]
[[simple:Diagonal]]
[[sk:Uhlopriečka]]
[[sl:Diagonala]]
[[th:เส้นทแยงมุม]]
[[uk:Діагональ]]
[[zh:對角線]]