### authors

- COAKLEY, EDWARD S.
- MARTINEZ DOPICO, FROILAN CESAR
- JOHNSON ., CHARLES ROYAL

- February 2008

- 796

- 813

- 4

- 428

- 0024-3795

- 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.