# Alternant matrix

Not to be confused with alternating sign matrix.

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

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

or more succinctly

${\displaystyle M_{i,j}=f_{j}(\alpha _{i})}$

for all indices i and j. (Some authors use the transpose of the above matrix.)

Examples of alternant matrices include Vandermonde matrices, for which ${\displaystyle f_{i}(\alpha )=\alpha ^{i-1}}$, and Moore matrices, for which ${\displaystyle f_{i}(\alpha )=\alpha ^{q^{i-1}}}$.

If ${\displaystyle n=m}$ and the ${\displaystyle f_{j}(x)}$ functions are all polynomials, there are some additional results: if ${\displaystyle \alpha _{i}=\alpha _{j}}$ for any ${\displaystyle i, then the determinant of any alternant matrix is zero (as a row is then repeated), thus ${\displaystyle (\alpha _{j}-\alpha _{i})}$ divides the determinant for all ${\displaystyle 1\leq i. As such, if one takes

${\displaystyle V={\begin{bmatrix}1&\alpha _{1}&\dots &\alpha _{1}^{n-1}\\1&\alpha _{2}&\dots &\alpha _{2}^{n-1}\\1&\alpha _{3}&\dots &\alpha _{3}^{n-1}\\\vdots &\vdots &\ddots &\vdots \\1&\alpha _{n}&\dots &\alpha _{n}^{n-1}\\\end{bmatrix}}}$

(a Vandermonde matrix), then ${\displaystyle \prod _{i divides such polynomial alternant determinants. The ratio ${\displaystyle {\frac {\det M}{\det V}}}$ is called a bialternant. The case where each function ${\displaystyle f_{j}(x)=x^{m_{j}}}$ forms the classical definition of the Schur polynomials.

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