Alternant matrix

Not to be confused with 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 $f_{1}(x)=\sin(x)$, $f_{2}(x)=\cos(x)$ 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 $-1\neq 0$. 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 $f_{1}(x)=\sin(x)$, $f_{2}=\cos(x)$ 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 ${\frac {A}{x+1}}+{\frac {B}{x+2}}={\frac {1}{(x+1)(x+2)}}$. 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 ${\begin{bmatrix}1/2&1/3&1/6\\1/3&1/4&1/12\\1/4&1/5&1/20\end{bmatrix}}\sim {\begin{bmatrix}1&0&1\\0&1&-1\\0&0&0\end{bmatrix}}$. 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 $A=1,B=-1$.

• If $n=m$ and $\alpha _{i}=\alpha _{j}$ for any $i\neq j$, 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 $1\leq i<j\leq n$. In particular, if V is a Vandermonde matrix, then ${\textstyle \prod _{i<j}(\alpha _{j}-\alpha _{i})=\det V}$ divides such polynomial alternant determinants. The ratio ${\textstyle {\frac {\det M}{\det V}}}$ 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

• 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.