Moore matrix

In linear algebra, a Moore determinant, introduced by E. H. Moore (1896), 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

$M={\begin{bmatrix}\alpha _{1}&\alpha _{1}^{q}&\dots &\alpha _{1}^{q^{n-1}}\\\alpha _{2}&\alpha _{2}^{q}&\dots &\alpha _{2}^{q^{n-1}}\\\alpha _{3}&\alpha _{3}^{q}&\dots &\alpha _{3}^{q^{n-1}}\\\vdots &\vdots &\ddots &\vdots \\\alpha _{m}&\alpha _{m}^{q}&\dots &\alpha _{m}^{q^{n-1}}\\\end{bmatrix}}$

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.

Dickson used the Moore determinant in finding the modular invariants of the general linear group over a finite field.

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.
• Moore, E. H. (1896), "A two-fold generalization of Fermat's theorem.", American M. S. Bull., 2: 189–199, doi:10.1090/S0002-9904-1896-00337-2, JFM 27.0139.05