NEAR-DIAGONAL REPRESENTATIVES OF CONJUGACY CLASSES OF ORTHOGONAL MATRICES

Abstract

Let F be an algebraically closed field of characteristic not 2. Define the orthogonal group, O n(F), as the subgroup of GL n(F) consisting of matrices X such that X-1 = X', the transpose of X. We prove that any A ∈ O n(F) having no elementary divisors (t ± 1)k, with k even, is conjugate to a direct sum of zigzag matrices. A zigzag matrix is a special kind of a band matrix with only five nonzero diagonals. We also propose a conjecture about the case excluded above.