Moore matrix

In linear algebra, a Moore matrix, introduced by E. H. Moore (1896), is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism applied to its columns (beginning with the zeroth power of the Frobenius automorphism in the first column), so it is 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, 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 are linearly dependent over the finite field of order q. So it is analogous to the Wronskian of several functions.

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

References

Dickson, Leonard Eugene (1958) [1901], Magnus, Wilhelm (ed.), Linear groups: With an exposition of the Galois field theory, Dover Phoenix editions, Dover Publications, ISBN 978-0-486-49548-4, MR 0104735, OCLC 52335047
David Goss (1996). "1. Additive Polynomials". Basic Structures of Function Field Arithmetic. Springer. pp. 1–33. doi:10.1007/978-3-642-61480-4_1. ISBN 3-540-63541-6.
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, JFM 27.0139.05

Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices

Fahrer / Team für die Suche eingeben:

Fahrer / Team für die Suche eingeben:

Langbahn Team – Weltmeisterschaft

Moore matrix

See also

References