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Diagonal
 
  A diagonal can refer to a line joining two nonconsecutive vertices of a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 or polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
, or in informal contexts any upward or downward sloping line. The word "diagonal" derives from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
 d?a?????? (diagonios), from dia- ("through", "across") and gonia ("angle", related to gony "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
 or cuboid
Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
,, and later adopted into Latin as diagonus ("slanting line").

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, in addition to its geometric meaning, a diagonal is also used in matrices to refer to a set of entries along a diagonal line.

Non-mathematical uses

In engineering
Engineering

Engineering is the discipline and profession of applying Technology and science knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and process that safely realize a desired objective and meet specified criteria....
, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding
Scaffolding

Scaffolding is a temporary framework used to support people and material in the construction or repair of buildings and other large structures....
) 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
Diagonal pliers

Diagonal pliers are pliers intended for the cutting of wire . They are sometimes called side cutting pliers or side cutters, although these terms are shared by other pliers designs and may lead to confusion....
 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 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
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
, a diagonal is a line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
 joining any two non-consecutive vertices. Therefore, a quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
 has two diagonals, joining opposite pairs of vertices. For any convex polygon
Convex polygon

In geometry, a polygon can be either convex or concave....
, all the diagonals are inside the polygon, but for re-entrant polygons, some diagonals are outside of the polygon.

Any n-sided polygon (n = 3), convex
Convex polygon

In geometry, a polygon can be either convex or concave....
 or concave, has diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or n − 3 diagonals.

Sides Diagonals
3 0
4 2
5 5
6 9
7 14
8 20
9 27
10 35
Sides Diagonals
11 44
12 54
13 65
14 77
15 90
16 104
17 119
18 135
Sides Diagonals
19 152
20 170
21 189
22 209
23 230
24 252
25 275
26 299
Sides Diagonals
27 324
28 350
29 377
30 405
31 434
32 464
33 495
34 527
Sides Diagonals
35 560
36 594
37 629
38 665
39 702
40 740
41 779
42 819


This is related to the Handshake Problem: In a room with n strangers, how many handshakes must take place so that everyone has met? The answer is found by adding the diagonals of an n-gon with the number of sides, n. Geometrically, the n strangers are the vertices of an n-gon, and the lines connecting the vertices (whether diagonals or sides) represent handshakes. Therefore, the total number of handshakes is given by:

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 a matrix with row index specified by and column index specified by , these would be elements with . For example, the identity matrix
Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere....
 can be defined as having entries of 1 on the main diagonal, and zeroes 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. In like manner to the above, superdiagonal elements can be specified by with . If otherwise unqualified, it refers to the one adjacent to the main diagonal. For example, the non-zero elements of the following matrix all lie in the superdiagonal:

Likewise, a subdiagonal entry is one that is directly below and to the left of the main diagonal. These can be specified as those elements with . The off-diagonal entries are those not on the main diagonal. A diagonal matrix
Diagonal matrix

In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero....
 is one whose off-diagonal entries are all zero. General matrix diagonals can be specified by an index measured relative to the main diagonal, the origin of the diagonal as it were, where , for which the superdiagonal has , subdiagonal, , and where in general those k-diagonal elements require that .

Geometry

By analogy, the subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of the Cartesian product
Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
 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
Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. That is to say, x is a fixed point of the function f if and only if f = x....
s of a mapping
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 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
Equivalence class

In mathematics, given a Set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
. This is related at a deep level with the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 and the zeros of vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
s. For example, the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 S1 has Betti number
Betti number

In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces....
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
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
 S1xS1 and observe that it can move off itself by the small motion (?, ?) to (?, ? + e). 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
    Jordan normal form

    In linear algebra, Jordan normal form shows that a given square matrix M over a field K containing the eigenvalues of M can be transformed into a certain normal form by changing the Basis ....
  • Main diagonal
    Main diagonal

    In linear algebra, the main diagonal of a matrix is the collection of cells where is equal to .The main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner....
  • Diagonal functor
    Diagonal functor

    In category theory, for any object a in any category C where the product a?a exists, there exists the diagonal morphismsatisfying...
  • Face diagonal
    Face diagonal

    In geometry, a face diagonal of a polyhedron is a diagonal on one of the faces , in contrast to a space diagonal passing through the interior of the polyhedron....
  • Space diagonal
    Space diagonal

    In a cuboid or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner....


External links

  • with interactive animation
  • from MathWorld
    MathWorld

    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
    .
  • of a matrix from MathWorld
    MathWorld

    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
    .
  • from Diagonal
    Diagonal

    A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
    .