Information AboutDiagonal |
| CATEGORIES ABOUT DIAGONAL | |
| elementary mathematics | |
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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 Polygon s. 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, : 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 s 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 Field s. For example the Circle ''S''1 has Betti Number s 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''1xS1; 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 |
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