alternant matrix

{{Distinguish|alternating sign 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 {{nowrap|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 {{nowrap|-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 {{nowrap|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 {{nowrap|\frac{A}{x+1} + \frac{B}{x+2} = \frac{1}{(x+1)(x+2)}.}} Choosing {{nowrap|f_1(x) = \frac{1}{x+1},}} {{nowrap|f_2(x) = \frac{1}{x+2},}} f_3(x) = \frac{1}{(x+1)(x+2)} and {{nowrap|(\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 {{nowrap|A = 1, B = -1.}}
  • If n = m and \alpha_i = \alpha_j for any {{nowrap|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 {{nowrap|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

See also

References

{{refbegin}}

  • {{cite book |first=Thomas |last=Muir | authorlink=Thomas Muir (mathematician) | title=A treatise on the theory of determinants |orig-date=1960 | publisher=Dover Publications | pages=321–363 |isbn=978-0-486-49553-8 |oclc=52203124 |date=2003}}
  • {{cite book |first=A.C. |last=Aitken | authorlink=Alexander Aitken | title=Determinants and Matrices | date=1956 | publisher=Oliver and Boyd Ltd | pages=111–123 |edition=9th |oclc=271302373}}
  • {{cite book |first=Richard P. |last=Stanley | authorlink=Richard P. Stanley | title=Enumerative Combinatorics | date=1999 | publisher=Cambridge University Press | pages=334–342 |edition=2nd |doi=10.1017/CBO9781139058520 |isbn=978-1-107-01542-5 |oclc=897778191}}

{{refend}}

{{Matrix classes}}

Category:Matrices (mathematics)

Category:Determinants