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 ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m\in X$, then the alternant matrix has size $m\times n$ and is defined by

$M=\left[\begin{array}{cccc}{f}_1({\alpha }_1)& f_2({\alpha }_1)& \cdots & f_n({\alpha }_1)\\ f_1({\alpha }_2)& f_2({\alpha }_2)& \cdots & f_n({\alpha }_2)\\ f_1({\alpha }_3)& f_2({\alpha }_3)& \cdots & f_n({\alpha }_3)\\ ⋮& ⋮& \ddots & ⋮\\ f_1({\alpha }_m)& f_2({\alpha }_m)& \cdots & f_n({\alpha }_m)\\ \end{array}\right]$

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 $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ and the alternant determinant is $-1\ne 0$. 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 $\left[\begin{array}{cc}0& 1\\ 0& -1\end{array}\right]$ 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 $\left[\begin{array}{ccc}1/2& 1/3& 1/6\\ 1/3& 1/4& 1/12\\ 1/4& 1/5& 1/20\end{array}\right]\sim \left[\begin{array}{ccc}1& 0& 1\\ 0& 1& -1\\ 0& 0& 0\end{array}\right]$. 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 $i\ne j$, 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 $1\le i<j\le n$. 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 ${\alpha }_1,\dots ,{\alpha }_m$ 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.

See also

• List of matrices
• Wronskian

References

• Thomas Muir (1960). A treatise on the theory of determinants. Dover Publications. pp. 321–363. 
• A. C. Aitken (1956). Determinants and Matrices. Oliver and Boyd Ltd. pp. 111–123. 
• Richard P. Stanley (1999). Enumerative Combinatorics. Cambridge University Press. pp. 334–342.