Invariant subspaces and Jordan form

This chapter starts by introducing the notion of root subspaces for quaternion matrices. These are basic invariant subspaces, and the chapter proves in particular that they enjoy the Lipschitz property with respect to perturbations of the matrix. Another important class of invariant subspaces are the one-dimensional ones, i.e., generated by eigenvectors. Existence of quaternion eigenvalues and eigenvectors is proved, which leads to the Schur triangularization theorem (in the context of quaternion matrices) and its many consequences familiar for real and complex matrices. Jordan canonical form for quaternion matrices is stated and proved (both the existence and uniqueness parts) in full detail. The chapter also discusses various concepts of determinants for square-size quaternion matrices. Several applications of the Jordan form are given, including functions of matrices and boundedness properties of systems of differential and difference equations with constant quaternion coefficients.