Diagonal

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In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and 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,

<math>d= \frac{n^2-3n}{2}.\, </math>

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 points of a 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 S¹ 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 S¹xS¹; 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 C where the product a×a exists, one may construct the diagonal morphism δ_a: a → a×a, satisfying π_kδ_a = id_a for k=1,2. The existence of this morphism is a consequence of the universal property which characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly to arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.

For concrete categories, the diagonal morphism can be simply described by its action on elements x of the object a. Namely, δ_a(x) = (x,x), the ordered pair formed from x. The reason for the name is that the graph of such a diagonal morphism is diagonal, for example the graph of the diagonal morphism R → R² on the real line is given by the line which is a graph of the equation y=x. The diagonal morphism into the infinite product X^∞ may provide an injection into the space of sequences valued in X; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions which the image of the diagonal map will fail to satisfy.

In particular, the category of categories has products, and so one finds the diagonal functor Δ: C → C×C given by Δ(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×b is a universal arrow from Δ 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 index category), for each object a in C, there is a constant functor Δ_a which maps each object j in J to a Δ_a(j) = a and maps each morphism j → k in J to the identity morphism on a. The diagonal functor Δ: C → C^J assigns to each object of C the constant functor at that object (Δ(a) = Δ_a ∈ C^J), and to each morphism f: a → b in C the obvious natural transformation in C^J (given by η_j = f). The the limit of any functor F: J → C is a universal arrow from Δ to F and a colimit is a universal arrow F → Δ.

The diagonal functor Δ is the left-adjoint of the product functor and the right-adjoint of the coproduct functor.

See also

• diagonal matrix
• main diagonalde:Diagonale (Geometrie)

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