Английская Википедия:Alternant matrix
Шаблон:Distinguish 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 <math>f_1, f_2, \dots, f_n</math> are functions from a set <math>X</math> to a field <math>F</math>, and <math>{\alpha_1, \alpha_2, \ldots, \alpha_m} \in X</math>, then the alternant matrix has size <math>m \times n</math> and is defined by
- <math>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}</math>
or, more compactly, <math>M_{ij} = f_j(\alpha_i)</math>. (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which <math>f_j(\alpha)=\alpha^{j-1}</math>, and Moore matrices, for which <math>f_j(\alpha)=\alpha^{q^{j-1}}</math>.
Properties
- The alternant can be used to check the linear independence of the functions <math>f_1, f_2, \dots, f_n</math> in function space. For example, let Шаблон:Nowrap <math>f_2(x) = \cos(x)</math> and choose <math>\alpha_1 = 0, \alpha_2 = \pi/2</math>. Then the alternant is the matrix <math>\left[\begin{smallmatrix}0 & 1 \\ 1 & 0 \end{smallmatrix}\right]</math> and the alternant determinant is Шаблон:Nowrap Therefore M is invertible and the vectors <math>\{\sin(x), \cos(x)\}</math> form a basis for their spanning set: in particular, <math>\sin(x)</math> and <math>\cos(x)</math> 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 Шаблон:Nowrap <math>f_2 = \cos(x)</math> and choose <math>\alpha_1 = 0, \alpha_2 = \pi</math>. Then the alternant is <math>\left[\begin{smallmatrix}0 & 1 \\ 0 & -1 \end{smallmatrix}\right]</math> and the alternant determinant is 0, but we have already seen that <math>\sin(x)</math> and <math>\cos(x)</math> 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 Шаблон:Nowrap Choosing Шаблон:Nowrap Шаблон:Nowrap <math>f_3(x) = \frac{1}{(x+1)(x+2)}</math> and Шаблон:Nowrap we obtain the alternant <math>\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}</math>. Therefore, <math>(1,-1,-1)</math> is in the nullspace of the matrix: that is, <math>f_1 - f_2 - f_3 = 0</math>. Moving <math>f_3</math> to the other side of the equation gives the partial fraction decomposition Шаблон:Nowrap
- If <math>n = m</math> and <math>\alpha_i = \alpha_j</math> for any Шаблон:Nowrap then the alternant determinant is zero (as a row is repeated).
- If <math>n = m</math> and the functions <math>f_j(x)</math> are all polynomials, then <math>(\alpha_j - \alpha_i)</math> divides the alternant determinant for all Шаблон:Nowrap In particular, if V is a Vandermonde matrix, then <math display="inline">\prod_{i < j} (\alpha_j - \alpha_i) = \det V</math> divides such polynomial alternant determinants. The ratio <math display="inline">\frac{\det M}{\det V}</math> is therefore a polynomial in <math>\alpha_1, \ldots, \alpha_m</math> called the bialternant. The Schur polynomial <math>s_{(\lambda_1, \dots, \lambda_n)}</math> is classically defined as the bialternant of the polynomials <math>f_j(x) = x^{\lambda_j}</math>.
Applications
- Alternant matrices are used in coding theory in the construction of alternant codes.
See also
References
