diagonaalsed

Diagonaalsed is an Estonian term referring to diagonals or diagonal-related properties in mathematics. The concept appears in both geometry, where it describes line segments connecting non-adjacent vertices of polygons, and in linear algebra, where it relates to the diagonal elements of matrices and to diagonalization.

In geometry, a diagonal of a polygon is a line segment that joins two vertices that are not connected by an edge. For a polygon with n vertices, the number of diagonals is n(n-3)/2. Convex polygons have all diagonals lying inside the polygon, whereas in concave polygons some diagonals may lie outside the boundary. Examples: a quadrilateral has two diagonals; a pentagon has five. Specific polygon types exhibit distinctive diagonal properties: in a rectangle, diagonals are equal in length and bisect each other; in a parallelogram they bisect each other without necessarily being equal; in a rhombus they are perpendicular and also bisect the angles.

In three dimensions, the term can extend to space diagonals, which connect opposite vertices of polyhedra. For a rectangular prism with side lengths l, w, and h, the space diagonal length is sqrt(l^2 + w^2 + h^2).

In linear algebra, diagonaalsed relates to diagonal elements of a square matrix. The main diagonal consists of entries a11, a22, ..., ann. A matrix is diagonal if all off-diagonal entries are zero. A matrix is diagonalizable if it is similar to a diagonal matrix, typically when there exists a basis of eigenvectors. Related concepts include the trace, which equals the sum of the diagonal elements, and the role of diagonals in simplifying powers and functions of matrices.