skewdiagonal

Skew diagonal, more commonly called the anti-diagonal or secondary diagonal, refers to a diagonal-like line of entries in a square matrix that runs from the top-right corner to the bottom-left corner. In an n-by-n matrix A, the skew diagonal consists of the entries a_{i, n+1−i} for i = 1, 2, …, n. This diagonal intersects the main diagonal only when n is odd, at the central element a_{(n+1)/2, (n+1)/2}.

A matrix that has nonzero entries only on its skew diagonal is called an anti-diagonal or skew-diagonal matrix. A standard example is the reversal matrix J, defined by J_{i,j} = 1 if i + j = n + 1 and J_{i,j} = 0 otherwise. The matrix J reverses the order of the standard basis vectors and represents a reflection about the vertical axis. The determinant of J is (-1)^{n(n−1)/2}, which is 1 for n ≡ 0 or 1 mod 4 and −1 for n ≡ 2 or 3 mod 4.

In practice, anti-diagonal elements appear in descriptions of palindromic structures, certain matrix decompositions, and linear transformations that reverse sequences. The anti-diagonal is distinct from the main diagonal, though it is related by symmetry: transposition moves the anti-diagonal into a position that pairs with the main diagonal in a specific pattern, and the overall arrangement of anti-diagonal entries often informs properties such as symmetry and trace in related contexts.

Example: a 4×4 anti-diagonal matrix with ones on the skew diagonal and zeros elsewhere is

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

Its determinant is 1, as det(J) = (-1)^{6}.