eidiagonaaliset

Eidiagonaaliset refers to the anti-diagonal elements of a square matrix. In an n-by-n matrix A = [aij], the eidiagonaaliset are the entries for which i + j = n + 1; equivalently, they lie on the reverse diagonal from the top-right to the bottom-left.

An anti-diagonal matrix is one in which all nonzero entries lie on this anti-diagonal. If the anti-diagonal elements are a1, a2, ..., an (from the top-right corner downward toward the bottom-left), the matrix is often written with zeros elsewhere. A convenient way to construct such a matrix is A = D J, where D = diag(a1, ..., an) is a diagonal matrix and J is the exchange matrix that has ones on its anti-diagonal and zeros elsewhere. This expresses the anti-diagonal structure as a product of a diagonal matrix and a reversal operation.

Key properties include that there are exactly n eidiagonaaliset elements in an n-by-n matrix, and the determinant of an anti-diagonal matrix equals det(A) = (-1)^{n(n-1)/2} ∏_{k=1}^n a_k, i.e., the product of the anti-diagonal entries up to a sign dependent on n. The anti-diagonal is closely related to symmetry operations that reverse the order of basis vectors and to processes that involve reversing data sequences.

Occurrences and usage appear in linear algebra problems, pattern recognition, and certain matrix factorizations where a matrix is arranged with nonzero values along the anti-diagonal. Understanding eidiagonaaliset helps in recognizing special matrix forms and in applying relevant determinant and transformation formulas.

See also: main diagonal, anti-diagonal, exchange matrix.