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
![{\displaystyle M={\begin{bmatrix}f_{1}(\alpha _{1})&f_{2}(\alpha _{1})&\cdots &f_{n}(\alpha _{1})\\f_{1}(\alpha _{2})&f_{2}(\alpha _{2})&\cdots &f_{n}(\alpha _{2})\\f_{1}(\alpha _{3})&f_{2}(\alpha _{3})&\cdots &f_{n}(\alpha _{3})\\\vdots &\vdots &\ddots &\vdots \\f_{1}(\alpha _{m})&f_{2}(\alpha _{m})&\cdots &f_{n}(\alpha _{m})\\\end{bmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f472562bd3e732cfab52fe9369604cc7bf402f5)
or, more compactly,
. (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which
, and Moore matrices, for which
.