Diagonal

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A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in some contexts any upward or downward sloping line. The word "diagonal" was originally from the Greek διαγωνιος (diagonios), used by both Strabo^[1] and Euclid^[2] to refer to a line connecting two vertices of a rhombus or cuboid,^[3] 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 matrices to refer to a set of entries along a diagonal line.

Contents 1 Non mathematical uses 2 Polygons 3 Matrices 4 Geometry 5 See also 6 External links 7 References

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 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, jncave]], has

<math>\frac{n^2-3n}{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 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. 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 points of a 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 fields. For example, the circle S¹ has Betti numbers 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 S¹xS¹ 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

• Diagonal matrix
• Jordan normal form
• Main diagonal
• Diagonal functor

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