Moore determinant

In linear algebra, a Moore matrix, named after E. H. Moore, is a determinant defined over a finite field from a square Moore matrix. A Moore matrix has successive powers of the Frobenius automorphism applied to the first column, i.e., an m &times; n matrix


 * $$M=\begin{bmatrix}

\alpha_1 & \alpha_1^q & \dots & \alpha_1^{q^{n-1}}\\ \alpha_2 & \alpha_2^q & \dots & \alpha_2^{q^{n-1}}\\ \alpha_3 & \alpha_3^q & \dots & \alpha_3^{q^{n-1}}\\ \vdots & \vdots & \ddots &\vdots \\ \alpha_m & \alpha_m^q & \dots & \alpha_m^{q^{n-1}}\\ \end{bmatrix}$$ or
 * $$M_{i,j} = \alpha_i^{q^{j-1}}$$

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

The Moore determinant of a square Moore matrix (so m=n) can be expressed as:


 * $$\det(V) = \prod_{\mathbf{c}} \left( c_1\alpha_1 + \cdots c_n\alpha_n \right), $$

where c runs over a complete set of direction vectors, made specific by having the last non-zero entry equal to 1.