Alternant matrix

In linear algebra, an alternant matrix is a matrix with a particular structure, in which successive columns have a particular function applied to their entries. An alternant determinant is the determinant of an alternant matrix. Such a matrix of size m × n may be written out as

$M=\begin{bmatrix} f_1(\alpha_1) & f_2(\alpha_1) & \dots & f_n(\alpha_1)\\ f_1(\alpha_2) & f_2(\alpha_2) & \dots & f_n(\alpha_2)\\ f_1(\alpha_3) & f_2(\alpha_3) & \dots & f_n(\alpha_3)\\ \vdots & \vdots & \ddots &\vdots \\ f_1(\alpha_m) & f_2(\alpha_m) & \dots & f_n(\alpha_m)\\ \end{bmatrix}$

or more succinctly

$M_{i,j} = f_j(\alpha_i)$

for all indices i and j. (Some authors use the transpose of the above matrix.)

Examples of alternant matrices include Vandermonde matrices, for which $f_i(\alpha)=\alpha^{i-1}$, and Moore matrices, for which $f_i(\alpha)=\alpha^{q^{i-1}}$.

If $n = m$ and the $f_j(x)$ functions are all polynomials, there are some additional results: if $\alpha_i = \alpha_j$ for any $i < j$, then the determinant of any alternant matrix is zero (as a row is then repeated), thus $(\alpha_j - \alpha_i)$ divides the determinant for all $1 \leq i < j \leq n$. As such, if one takes

$V = \begin{bmatrix} 1 & \alpha_1 & \dots & \alpha_1^{n-1} \\ 1 & \alpha_2 & \dots & \alpha_2^{n-1} \\ 1 & \alpha_3 & \dots & \alpha_3^{n-1} \\ \vdots & \vdots & \ddots &\vdots \\ 1 & \alpha_n & \dots & \alpha_n^{n-1} \\ \end{bmatrix}$

(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 called a bialternant. The case where each function $f_j(x) = x^{m_j}$ forms the classical definition of the Schur polynomials.

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

See also

• List of matrices

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.