skevadiagonala

Skevadiagnala, commonly called the secondary diagonal or anti-diagonal, refers to the diagonal that runs from the top-right corner to the bottom-left corner of a square matrix. In an n x n matrix A, the skevadiagonala consists of the elements a_{i, n+1−i} for i = 1,...,n. It is distinct from the main diagonal, which consists of a_{i,i}.

In non-square matrices, the anti-diagonal can be defined as the set of entries with i + j = n + 1, yielding a segment of length min(m,n) in an m x n matrix. The concept is often used in matrix transformations and operations that involve reversing the order of rows or columns.

Matrices with nonzero entries only on the skevadiagonala are called anti-diagonal matrices. The standard anti-diagonal matrix, sometimes referred to as the exchange matrix J, has ones on the anti-diagonal and zeros elsewhere. Multiplication by J reverses the order of columns or rows, depending on the side of multiplication, and is used to implement row or column reversals.

Properties and applications: The skevadiagonala appears in analyses of palindromic and skew-symmetric structures and in certain determinant and eigenvalue computations. For an anti-diagonal matrix with anti-diagonal elements a_i, the determinant is (-1)^{n(n−1)/2} times the product of the a_i when the matrix is n x n.

Examples: For A = [[1,2,3],[4,5,6],[7,8,9]], the skevadiagonal consists of the elements 3, 5, and 7 (positions (1,3), (2,2), and (3,1)).