On the decomposition of orthogonalities into symmetries
Peter Scherk · Proceedings of the American Mathematical Society · 1950
Two vectors a and b are called perpendicular if a'Gb=O. Two subspaces 9Z* and 9** are perpendicular if x'Gy =0 for all xC9Z*, yCgz**. Obviously, these relations are symmetric. The vectors perpendicular to a given vector respectively to a given m-space form an (n-1)space, respectively (n-m)-space. We call the matrix T orthogonal if it leaves the expression x'Gy unchanged for all x and y. This condition is equivalent to