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.)
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.