Matrices with Orthogonal Groups Admitting only Determinant One Articles uri icon

publication date

  • February 2008

start page

  • 796

end page

  • 813


  • 4


  • 428

International Standard Serial Number (ISSN)

  • 0024-3795

Electronic International Standard Serial Number (EISSN)

  • 1873-1856


  • The K-Orthogonal group of an n-by-n matrix K is defined as the set of nonsingular n-by-n matrices Asatisfying ATKA=K, where the superscript T denotes transposition. These form a group under matrix multiplication. It is well-known that if K is skew-symmetric and nonsingular the determinant of every element of the K-Orthogonal group is +1, i.e., the determinant of any symplectic matrix is +1. We present necessary and sufficient conditions on a real or complex matrix K so that all elements of the K-Orthogonal group have determinant +1. These necessary and sufficient conditions can be simply stated in terms of the symmetric and skew-symmetric parts of K, denoted by Ks and Kw respectively, as follows: the determinant of every element in the K-Orthogonal group is +1 if and only if the matrix pencil Kw-lambdaKs is regular and the matrix (Kw-lambda0Ks)-1Kw has no Jordan blocks associated to the zero eigenvalue with odd dimension, where lambda0 is any number such that det(Kw-lambda0Ks)≠0.