In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternant matrix.
Generally, if are functions from a set to a field , and , then the alternant matrix has size and is defined by
The alternant can be used to check the linear independence of the functions in function space. For example, let and choose . Then the alternant is the matrix and the alternant determinant is . Therefore M is invertible and the vectors form a basis for their spanning set: in particular, and are linearly independent.
Linear dependence of the columns of an alternant does not imply that the functions are linearly dependent in function space. For example, let and choose . Then the alternant is and the alternant determinant is 0, but we have already seen that and are linearly independent.
Despite this, the alternant can be used to find a linear dependence if it is already known that one exists. For example, we know from the theory of partial fractions that there are real numbers A and B for which Choosing and , we obtain the alternant Therefore is in the nullspace of the matrix: that is, . Moving to the other side of the equation gives the partial fraction decomposition .
If and for any , then the alternant determinant is zero (as a row is repeated).
If and the functions are all polynomials, then divides the alternant determinant for all . In particular, if V is a Vandermonde matrix, then divides such polynomial alternant determinants. The ratio is therefore a polynomial in called the bialternant. The Schur polynomial is classically defined as the bialternant of the polynomials .