In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functionss pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternant matrix.

to a field and is defined by

. (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, and Moore matrices.

Properties

• The alternant can be used to check the linear independence in function space. For example, let and the alternant determinant is Therefore M 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 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 is in the nullspace to the other side of the equation gives the partial fraction decomposition

  for any then the alternant determinant is zero (as a row is repeated).

  are all polynomials divides the alternant determinant for all In particular, if V is a Vandermonde matrix called the bialternant. The Schur polynomial.

Applications

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

See also

• List of matrices

• Wronskian

References