Diagonalnych

Diagonalnych is the genitive plural form of the Polish adjective diagonalny, used to describe things related to diagonals. In practice it appears in phrases where the noun is in the genitive plural, for example elementów diagonalnych (diagonal elements), macierzy diagonalnych (diagonal matrices) or długości przekątnych (lengths of diagonals). The form is common in technical and mathematical language to indicate diagonal-related properties.

In geometry, a diagonal is a line segment connecting non-adjacent vertices of a polygon. Diagonals play a central role in understanding shapes such as quadrilaterals, where every convex quadrilateral has exactly two diagonals that intersect in general position. In three-dimensional figures, one can also consider space diagonals, which connect opposite vertices of polyhedra. In linear algebra, diagonal elements of a matrix are those with equal row and column indices, and matrices that have nonzero entries only on this main diagonal are called diagonal matrices. The concept extends to related notions such as the anti-diagonal, where nonzero elements lie on the opposite diagonal (i + j = n + 1 for an n-by-n matrix).

Grammatically, diagonalnych is a plural genitive form used with masculine and mixed noun phrases in contexts like 'elementów diagonalnych' or 'macierzy diagonalnych.' It can also appear in phrases describing diagonal properties, such as 'własność diagonalności' (diagonality). Although the form is inflected, its core meaning remains tied to diagonals or diagonal-related concepts across mathematics, computer science, and design.