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 ${\displaystyle f_{1},f_{2},\dots ,f_{n}}${\displaystyle f_{1},f_{2},\dots ,f_{n}} are functions from a set ${\displaystyle X}${\displaystyle X} to a field ${\displaystyle F}${\displaystyle F}, and ${\displaystyle {\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}\in X}${\displaystyle {\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}\in X}, then the alternant matrix has size ${\displaystyle m\times n}${\displaystyle m\times n} and is defined by

${\displaystyle M={\begin{bmatrix}f_{1}(\alpha _{1})&f_{2}(\alpha _{1})&\cdots &f_{n}(\alpha _{1})\\f_{1}(\alpha _{2})&f_{2}(\alpha _{2})&\cdots &f_{n}(\alpha _{2})\\f_{1}(\alpha _{3})&f_{2}(\alpha _{3})&\cdots &f_{n}(\alpha _{3})\\\vdots &\vdots &\ddots &\vdots \\f_{1}(\alpha _{m})&f_{2}(\alpha _{m})&\cdots &f_{n}(\alpha _{m})\\\end{bmatrix}}}${\displaystyle 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, ${\displaystyle M_{ij}=f_{j}(\alpha _{i})}${\displaystyle M_{ij}=f_{j}(\alpha _{i})}. (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which ${\displaystyle f_{j}(\alpha )=\alpha ^{j-1}}${\displaystyle f_{j}(\alpha )=\alpha ^{j-1}}, and Moore matrices, for which ${\displaystyle f_{j}(\alpha )=\alpha ^{q^{j-1}}}${\displaystyle f_{j}(\alpha )=\alpha ^{q^{j-1}}}.

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

• If ${\displaystyle n=m}${\displaystyle n=m} and ${\displaystyle \alpha _{i}=\alpha _{j}}${\displaystyle \alpha _{i}=\alpha _{j}} for any ${\displaystyle i\neq j}${\displaystyle i\neq j}, then the alternant determinant is zero (as a row is repeated).

• If ${\displaystyle n=m}${\displaystyle n=m} and the functions ${\displaystyle f_{j}(x)}${\displaystyle f_{j}(x)} are all polynomials, then ${\displaystyle (\alpha _{j}-\alpha _{i})}${\displaystyle (\alpha _{j}-\alpha _{i})} divides the alternant determinant for all ${\displaystyle 1\leq i<j\leq n}${\displaystyle 1\leq i<j\leq n}. In particular, if V is a Vandermonde matrix, then ${\textstyle \prod _{i<j}(\alpha _{j}-\alpha _{i})=\det V}${\textstyle \prod _{i<j}(\alpha _{j}-\alpha _{i})=\det V} divides such polynomial alternant determinants. The ratio ${\textstyle {\frac {\det M}{\det V}}}${\textstyle {\frac {\det M}{\det V}}} is therefore a polynomial in ${\displaystyle \alpha _{1},\ldots ,\alpha _{m}}${\displaystyle \alpha _{1},\ldots ,\alpha _{m}} called the bialternant. The Schur polynomial ${\displaystyle s_{(\lambda _{1},\dots ,\lambda _{n})}}${\displaystyle s_{(\lambda _{1},\dots ,\lambda _{n})}} is classically defined as the bialternant of the polynomials ${\displaystyle f_{j}(x)=x^{\lambda _{j}}}${\displaystyle f_{j}(x)=x^{\lambda _{j}}}.

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

• List of matrices
• Wronskian

• Matrices (mathematics)
• Determinants