definition only requires some invertibility, so one can substitute for Q any matrix M whose eigenvalues do not include −1. For example, Note that A is skew-symmetric (respectively, skew-Hermitian) if and only if Q is orthogonal (respectively, unitary) with no eigenvalue −1. Operator map An infinite-dimensional version of an inner product space is a Hilbert space, and one can no longer speak of matrices. However, matrices are merely representations of linear operators, and these can be used. So, generalizing both the matrix mapping and the complex plane mapping, one may define a Cayley transform of operators. Here the domain of U, dom U, is (A+iI) dom A. See self-adjoint operator for further details. See also Bilinear transform Extensions of symmetric operators References Sterling K. Berberian (1974) Lectures in Functional Analysis and Operator Theory, Graduate Texts in Mathematics #15, pages 278, 281, Springer-Verlag ; reprinted as article 52 (pp. 332–336) in Lokenath Debnath & Piotr Mikusiński (1990) Introduction to Hilbert Spaces with Applications, page 213, Academic Press Gilbert Helmberg (1969) Introduction to Spectral Theory in Hilbert Space, page 288, § 38: The Cayley Transform, Applied Mathematics and Mechanics #6, North Holland Henry Ricardo (2010) A Modern Introduction to Linear Algebra, page 504, CRC Press . External links ; translated from the Russian Conformal mappings Transforms ru:Преобразование Мёбиуса#Примеры