Alternant matrix explained

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,} ... , 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 x n and is defined byM = \begin{bmatrix} f _{1(\alpha} _{1)} & f _{2(\alpha} _{1)} & … & f _{n(\alpha} _{1)\\ f} _{1(\alpha} _{2)} & f _{2(\alpha} _{2)} & … & f _{n(\alpha} _{2)\\ f} _{1(\alpha} _{3)} & f _{2(\alpha} _{3)} & … & f _{n(\alpha} _{3)\\ \vdots} & \vdots & \ddots &\vdots \\ f _{1(\alpha} _{m)} & f _{2(\alpha} _{m)} & … & 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 -1} _{j(\alpha) = \alpha} , and Moore matrices, for which f ^{q ^{j-1}} _{j(\alpha) = \alpha} .

Properties

• The alternant can be used to check the linear independence of the functions

f _{1,} f _{2,} ... , f _{n} in function space. For example, let 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 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 _{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 Choosing

f _{3(x)} = (1)/((x +1 )(x +2 )) and 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

• If

n = m and \alpha _{i} = \alpha _{j} for any 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 In particular, if V is a Vandermonde matrix, then $\backslash prod\_\; (\backslash alpha\_j\; -\; \backslash alpha\_i)\; =\; \backslash det\; V$ divides such polynomial alternant determinants. The ratio $\backslash frac$ is therefore a polynomial in \alpha _{1,} \ldots, \alpha _{m} called the bialternant. The Schur polynomial s _{( λ _{1,} ... , λ _{n)}} is classically defined as the bialternant of the polynomials f _{j(x)} = x ^{λ _{j}} .

Applications

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

See also

• List of matrices
• Wronskian

References

• Book: Thomas Muir (mathematician)

. Thomas Muir . Thomas Muir (mathematician) . A treatise on the theory of determinants . 1960 . . 321–363 .

• Book: Alexander Aitken

. A. C. Aitken . Alexander Aitken . Determinants and Matrices . 1956 . Oliver and Boyd Ltd . 111–123 .

• Book: Richard P. Stanley

. Richard P. Stanley . Richard P. Stanley . Enumerative Combinatorics . 1999 . . 334–342 .