Alternant matrix

Not to be confused with alternating sign 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.

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.

Alternant matrix

See also

References