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A 'diagonal' can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in 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

Non mathematical uses

Polygons

Matrices

Geometry

See also

External links

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, joining opposite pairs of vertices. For any convex polygon, 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), even convex or concave, has
:
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 ''S''¹ 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 ''S''¹xS¹ 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

★ Face diagonal

★ Space diagonal

External links

★ Diagonals of a polygon with interactive animation

★ Polygon diagonal from MathWorld.

★ Diagonal of a matrix from MathWorld.

References

1. Strabo, Geography 2.1.36-37
2. Euclid, Elements book 11, proposition 28
3. Euclid, Elements book 11, proposition 38