nLab
Cayley transform

Contents

Idea

The classical Cayley transform is an involutive bijection between the vector space of skew-symmetric matrices to the group of special orthogonal matrices, given by the formula

XI+XIX. X \mapsto \frac{I + X}{I - X} \,.

There are many generalizations (some non-involutive), say for J-orthogonal matrices, to shifted center, for operators, in complex analysis and so on.

Literature

The general case related to symmetric spaces: