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Diagonal

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In mathematics, '''diagonal''' has a geometric meaning, and a derived meaning as used in square tables and matrix (math) matrix terminology.

Polygons
As applied to a polygon, a '''diagonal''' is a line segment joining two vertices that are not adjacent. Therefore a quadrilateral has two diagonals, joining opposite pairs of vertices. For a convex polygon the diagonals run inside the polygon. This is not so for re-entrant polygons. In fact a polygon is convex if and only if the diagonals are internal. When ''n'' is the number of vertices in a polygon and ''d'' is the number of possible different diagonals, each vertex has possible diagonals to all other vertices save for itself and the two adjacent vertices, or ''n''-3 diagonals; this multiplied by the number of vertices is :(''n'' − 3) × ''n'', which counts each diagonal twice (once for each vertex) — therefore, :

d= \frac{n^2-3n}{2}.\,

Matrices
In the case of a square matrix, the ''main'' or ''principal diagonal'' is the diagonal line of entries running north-west to south-east. For example the identity matrix can be described as having entries 1 on main diagonal, and 0 elsewhere. The north-east to south-west diagonal is sometimes described as the ''minor'' diagonal. A ''superdiagonal'' entry would be one that is above, and to the right of, 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'''. It is the graph of the identity relation. It plays an important part in geometry: for example the fixed point (mathematics) fixed points of a function (mathematics) mapping ''F'' from ''X'' to itself may be obtained by intersecting the graph of ''F'' with the diagonal. Quite a major role is played in geometric studies by the idea of intersecting the diagonal ''with itself'': not directly, but by perturbing it within an equivalence class. This is related at quite a deep level with the Euler characteristic and the zeroes of vector fields. For example the circle ''S''¹ has Betti numbers 1, 1, 0, 0, 0, ... and so Euler characteristic 0. A geometric way of saying that is to look at the diagonal on the two-torus ''S''¹xS¹; and to 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.

Category theory
In category theory, for any object ''a'' in any category (mathematics) category ''C'' where the product (category theory) product ''a''Ã—''a'' exists, one may construct the '''diagonal morphism''' Î´_''a'': ''a'' â†’ ''a''Ã—''a'', satisfying Ï€_''k''Î´_''a'' = id_''a'' for ''k''=1,2. The existence of this morphism is a consequence of the universal property which characterization (mathematics) characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly to arbitrary products. For concrete categories, the diagonal morphism can be simply described by its action on elements ''x'' of the object ''a''. Namely, Î´_''a''(''x'') = (''x'',''x''), the ordered pair formed from ''x''. The reason for the name is that the graph of a function graph of such a diagonal morphism is diagonal, for example the graph of the diagonal morphism ''R'' â†’ ''R''² is given by the line which is a graph of the equation ''y''=''x''. In particular, the category of categories has products, and so one finds the '''diagonal functor''' Î”: ''C'' â†’ ''C''Ã—''C'' given by Î”(''a'') = (''a'',''a''), the ordered pair for any object ''a'' in ''C''. This functor can be employed to give a succinct alternate description of the product of objects ''within'' the category ''C'': a product ''a''Ã—''b'' is a universal arrow from Î” to (''a'',''b''). The arrow comprises the projection maps. More generally, in any functor category ''C''^''J'' (here ''J'' should be thought of as a small category small index category), for each object ''a'' in ''C'', there is a constant functor Î”_''a'' which maps each object ''j'' in ''J'' to ''a'' Î”_''a''(''j'') = ''a'' and maps each morphism ''j'' â†’ ''k'' in ''J'' to the identity morphism on ''a''. The diagonal functor Î”: ''C'' â†’ ''C''^''J'' assigns to each object of ''C'' the constant functor at that object (Î”(''a'') = Î”_''a'' âˆˆ ''C''^''J''), and to each morphism ''f'': ''a'' â†’ ''b'' in ''C'' the obvious natural transformation in ''C''^''J'' (given by Î·_''j'' = ''f''). The the limit (category theory) limit of any functor ''F'': ''J'' â†’ ''C'' is a universal arrow from Î” to ''F'' and a colimit is a universal arrow ''F'' â†’ Î”. The diagonal functor Î” is the adjoint functors left-adjoint of the product (category theory) product functor and the right-adjoint of the coproduct coproduct functor.

See also
* diagonal matrix * main diagonal Category:Elementary mathematics de:Diagonale (Geometrie) es:Diagonal fr:Diagonale it:Diagonale he:×?×œ×›×¡×•×Ÿ hu:Ã?tlÃ³ nl:Diagonaal pl:PrzekÄ…tna pt:Diagonal simple:Diagonal sl:Diagonala sv:Diagonal zh:å°?è§’ç·š

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