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 × n matrix

[math]\displaystyle{ 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} }[/math]

or

[math]\displaystyle{ M_{i,j} = \alpha_i^{q^{j-1}} }[/math]

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:

[math]\displaystyle{ \det(V) = \prod_{\mathbf{c}} \left( c_1\alpha_1 + \cdots c_n\alpha_n \right), }[/math]

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

See also

• Alternant matrix
• Vandermonde determinant
• List of matrices

References

• David Goss (1996). Basic Structures of Function Field Arithmetic. Springer Verlag. ISBN 3-540-63541-6.  Chapter 1.