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 $$f_1, f_2, \dots, f_n$$ are functions from a set $$X$$ to a field $$F$$, and $${\alpha_1, \alpha_2, \ldots, \alpha_m} \in X$$, then the alternant matrix has size $$m \times n$$ and is defined by
 * $$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, $$M_{ij} = f_j(\alpha_i)$$. (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which $$f_j(\alpha)=\alpha^{j-1}$$, and Moore matrices, for which $$f_j(\alpha)=\alpha^{q^{j-1}}$$.

Properties

 * The alternant can be used to check the linear independence of the functions $$f_1, f_2, \dots, f_n$$ in function space. For example, let $f_1(x) = \sin(x)$, $$f_2(x) = \cos(x)$$ and choose $$\alpha_1 = 0, \alpha_2 = \pi/2$$. Then the alternant is the matrix $$\left[\begin{smallmatrix}0 & 1 \\ 1 & 0 \end{smallmatrix}\right]$$ and the alternant determinant is $-1 \neq 0$. Therefore M is invertible and the vectors $$\{\sin(x), \cos(x)\}$$ form a basis for their spanning set: in particular, $$\sin(x)$$ and $$\cos(x)$$ 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 $f_1(x) = \sin(x)$, $$f_2 = \cos(x)$$ and choose $$\alpha_1 = 0, \alpha_2 = \pi$$. Then the alternant is $$\left[\begin{smallmatrix}0 & 1 \\ 0 & -1 \end{smallmatrix}\right]$$ and the alternant determinant is 0, but we have already seen that $$\sin(x)$$ and $$\cos(x)$$ 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 $\frac{A}{x+1} + \frac{B}{x+2} = \frac{1}{(x+1)(x+2)}$. Choosing $f_1(x) = \frac{1}{x+1}$, $f_2(x) = \frac{1}{x+2}$, $$f_3(x) = \frac{1}{(x+1)(x+2)}$$ and $(\alpha_1,\alpha_2,\alpha_3) = (1,2,3)$, we obtain the alternant $$\begin{bmatrix} 1/2 & 1/3 & 1/6 \\ 1/3 & 1/4 & 1/12 \\ 1/4 & 1/5 & 1/20 \end{bmatrix} \sim \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{bmatrix}$$. Therefore, $$(1,-1,-1)$$ is in the nullspace of the matrix: that is, $$f_1 - f_2 - f_3 = 0$$. Moving $$f_3$$ to the other side of the equation gives the partial fraction decomposition $A = 1, B = -1$.


 * If $$n = m$$ and $$\alpha_i = \alpha_j$$ for any $i \neq j$, then the alternant determinant is zero (as a row is repeated).


 * If $$n = m$$ and the functions $$f_j(x)$$ are all polynomials, then $$(\alpha_j - \alpha_i)$$ divides the alternant determinant for all $1 \leq i < j \leq n$. In particular, if V is a Vandermonde matrix, then $\prod_{i < j} (\alpha_j - \alpha_i) = \det V$ divides such polynomial alternant determinants. The ratio $\frac{\det M}{\det V}$  is therefore a polynomial in $$\alpha_1, \ldots, \alpha_m$$ called the bialternant. The Schur polynomial $$s_{(\lambda_1, \dots, \lambda_n)}$$ is classically defined as the bialternant of the polynomials $$f_j(x) = x^{\lambda_j}$$.

Applications

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