Matrices with orthogonal groups admitting only determinant one Original Research Article

The K-Orthogonal group of an n-by-n matrix K is defined as the set of nonsingular n-by-n matrices A satisfying 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-λKs is regular and the matrix (Kw-λ0Ks)-1Kw has no Jordan blocks associated to the zero eigenvalue with odd dimension, where λ0 is any number such that det(Kw-λ0Ks)≠0.