Alternant matrix

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Not to be confused with alternating sign matrix.

In linear algebra, an alternant matrix is a matrix with a particular structure, in which successive columns have a particular function applied to their entries. An alternant determinant is the determinant of an alternant matrix. Such a matrix of size m × n may be written out as

or more succinctly

for all indices i and j. (Some authors use the transpose of the above matrix.)

Examples of alternant matrices include Vandermonde matrices, for which , and Moore matrices, for which .

If and the functions are all polynomials, there are some additional results: if for any , then the determinant of any alternant matrix is zero (as a row is then repeated), thus divides the determinant for all . As such, if one takes

(a Vandermonde matrix), then divides such polynomial alternant determinants. The ratio is called a bialternant. The case where each function forms the classical definition of the Schur polynomials.

Alternant matrices are used in coding theory in the construction of alternant codes.

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