antidiagonaal

An antidiagonal, or anti-diagonal, is the diagonal of a square matrix that runs from the top-right corner to the bottom-left corner. In an n×n matrix A = [aij], the antidiagonal consists of the entries ai, n+1−i for i = 1,…,n. Equivalently, these positions satisfy i + j = n + 1, contrasting with the main diagonal, where i = j.

An anti-diagonal matrix is a matrix in which all entries outside the antidiagonal are zero. The most notable example is the reverse-identity matrix Jn, which has ones on the antidiagonal and zeros elsewhere. Jn is a permutation matrix that reverses the order of the standard basis vectors.

If an n×n matrix A has nonzero entries only on its antidiagonal, write A with a1, a2, …, an on the antidiagonal from the top-right to the bottom-left. Then A can be expressed as A = D · Jn, where D = diag(a1, a2, …, an). Consequently, the determinant of A is det(A) = det(D) det(Jn) = (a1 a2 … an) · (−1)^{n(n−1)/2}.

In rectangular matrices, the term is sometimes generalized to the line running from the top-right to the bottom-left that contains min(m, n) elements, though the standard focus is on square matrices.

Applications of antidiagonal structure appear in certain linear systems and in algorithms for structured matrices, where exploiting the anti-diagonal arrangement can simplify computations and storage. See also main diagonal, secondary/antidiagonal concepts in matrix theory, and permutation matrices.