2021/12/14 by Rod Gow, Gow, Rod
Computer Science · Mathematics · #15A03 #15A33 #Advanced Topics in Algebra #Coding theory and cryptography #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2112.07293
openalex publication_date 2021/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a field and let Mn(K) denote the space of n x n matrices with entries in K. Let M be a subspace of Mn(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the determinantal polynomial PM of M with respect to the basis. It is a homogeneous polynomial of degree n in d indeterminates that gives the determinant of any element of M by evaluation in Kd. This paper investigates the interrelationship of M and PM. We confine ourselves to finite fields K, where we can obtain useful information by applying the Lang-Weil theorem on the number of zeros of absolutely irreducible polynomials. A combination of Chevalley's theorem on the zeros of polynomials in several variables and the Lang-Weil theorem leads to theorems about the characteristic polynomials of elements of M when n is a prime. We also draw attention to cases when the elements of M with non-zero determinant are a proper subspace, and provide non-trivial examples of this phenomenon.