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

$M=\left[\begin{array}{cccc}{\alpha }_1& {\alpha }_1^q& \dots & {\alpha }_1^{q^{n-1}}\\ {\alpha }_2& {\alpha }_2^q& \dots & {\alpha }_2^{q^{n-1}}\\ {\alpha }_3& {\alpha }_3^q& \dots & {\alpha }_3^{q^{n-1}}\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_m& {\alpha }_m^q& \dots & {\alpha }_m^{q^{n-1}}\\ \end{array}\right]$

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:

$\det (V)=\prod _{\mathbf{𝐜}}(c_1{\alpha }_1+\cdots c_n{\alpha }_n),$

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.