Definition:Alternant (Linear Algebra)

This page is about Alternant in the context of Linear Algebra. For other uses, see Alternant.

Definition

An alternant is a determinant of order $n$ such that the element in the $i$th row and $j$th column is defined as:

$\map {f_i} {r_j}$

where:

the $f_i$ are $n$ mappings

the $r_j$ are $n$ elements.

An alternant can also be defined as a determinant of order $n$ such that the element in the $i$th row and $j$th column is defined as:

$\map {f_j} {r_i}$

where:

the $f_j$ are $n$ mappings

the $r_i$ are $n$ elements.

Hence, such as to be the transpose of the alternant as defined.

An example of an alternant:

$\begin {vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end {vmatrix}$

Here we have:

$r_1$, $r_2$ and $r_3$ are identified with $x$, $y$ and $z$

$f_1$ is identified with the constant mapping: $\map {f_1} {x_i} = 1$

$f_2$ is identified with the identity mapping: $\map {f_2} {x_i} = x_i$

$f_3$ is identified with the square function: $\map {f_3} {x_i} = {x_i}^2$