Moore matrix

Inlinear algebra,aMoore matrix,introduced byE. H. Moore(1896), is amatrixdefined over afinite field.When it is a square matrix itsdeterminantis called aMoore determinant(this is unrelated to theMoore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of theFrobenius automorphismapplied to its columns (beginning with the zeroth power of the Frobenius automorphism in the first column), so it is anm×nmatrix $$M={\begin{bmatrix}\alpha _{1}&\alpha _{1}^{q}&\dots &\alpha _{1}^{q^{n-1}}\\\alpha _{2}&\alpha _{2}^{q}&\dots &\alpha _{2}^{q^{n-1}}\\\alpha _{3}&\alpha _{3}^{q}&\dots &\alpha _{3}^{q^{n-1}}\\\vdots &\vdots &\ddots &\vdots \\\alpha _{m}&\alpha _{m}^{q}&\dots &\alpha _{m}^{q^{n-1}}\\\end{bmatrix}}$$ or $$M_{i,j}=\alpha _{i}^{q^{j-1}}$$ for all indicesiandj.(Some authors use thetransposeof the above matrix.)

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

$$\det(V)=\prod _{\mathbf {c} }\left(c_{1}\alpha _{1}+\cdots +c_{n}\alpha _{n}\right),$$

wherecruns over a complete set of direction vectors, made specific by having the last non-zero entry equal to 1, i.e.,

$$\det(V)=\prod _{1\leq i\leq n}\prod _{c_{1},\dots,c_{i-1}}\left(c_{1}\alpha _{1}+\cdots +c_{i-1}\alpha _{i-1}+\alpha _{i}\right).$$

In particular the Moore determinant vanishes if and only if the elements in the left hand column arelinearly dependentover the finite field of orderq.So it is analogous to theWronskianof several functions.

Dickson used the Moore determinant in finding themodular invariantsof thegeneral linear groupover a finite field.

See also

• Alternant matrix
• Vandermonde matrix
• Vandermonde determinant
• List of matrices

References

• Dickson, Leonard Eugene(1958) [1901],Magnus, Wilhelm(ed.),Linear groups: With an exposition of the Galois field theory,Dover Phoenix editions, New York:Dover Publications,ISBN978-0-486-49548-4,MR0104735
• David Goss(1996).Basic Structures of Function Field Arithmetic.Springer Verlag.ISBN3-540-63541-6.Chapter 1.
• Moore, E. H.(1896), "A two-fold generalization of Fermat's theorem.",Bulletin of the American Mathematical Society,2(7): 189–199,doi:10.1090/S0002-9904-1896-00337-2,JFM27.0139.05