# Diagonal

In mathematics, **diagonal** has a geometric meaning, and a derived meaning as used in square tables and matrix terminology.

## Contents

## 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,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 geomet ry: 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 *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*^{1}xS^{1}; 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.

## See also

de:Diagonale (Geometrie) es:Diagonal it:Diagonale he:אלכסון hu:Átló nl:Diagonaal pl:Przekątna pt:Diagonal simple:Diagonal sv:Diagonal zh:對角線