Alternant matrix

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 ${\displaystyle f_{1},f_{2},\dots ,f_{n}}$ are functions from a set ${\displaystyle X}$ to a field ${\displaystyle F}$, and ${\displaystyle {\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}\in X}$, then the alternant matrix has size ${\displaystyle m\times n}$ and is defined by

${\displaystyle M={\begin{bmatrix}f_{1}(\alpha _{1})&f_{2}(\alpha _{1})&\cdots &f_{n}(\alpha _{1})\\f_{1}(\alpha _{2})&f_{2}(\alpha _{2})&\cdots &f_{n}(\alpha _{2})\\f_{1}(\alpha _{3})&f_{2}(\alpha _{3})&\cdots &f_{n}(\alpha _{3})\\\vdots &\vdots &\ddots &\vdots \\f_{1}(\alpha _{m})&f_{2}(\alpha _{m})&\cdots &f_{n}(\alpha _{m})\\\end{bmatrix}}}$

or, more compactly, ${\displaystyle M_{ij}=f_{j}(\alpha _{i})}$. (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which ${\displaystyle f_{j}(\alpha )=\alpha ^{j-1}}$, and Moore matrices, for which ${\displaystyle f_{j}(\alpha )=\alpha ^{q^{j-1}}}$.