Classes of Linear Transformations

Abstract

The classical development of linear algebra in the context of definite inner product spaces includes some central questions concerning linear transformations and matrices with symmetry properties. These properties arise in natural ways in the many applications of linear algebra in physics, statistics, mechanics, and so on. The most important of these symmetries include the selfadjoint and unitary properties.

In a similar way, applications requiring analysis on spaces with indefinite inner products also lead to the study of linear transformations and matrices with analogous symmetries. The most common of these are, once more, the (suitably defined) properties of selfadjoint and unitary transformations and their matrix representations. This chapter develops these fundamental ideas in the setting of complex matrices acting on Cⁿ with an indefinite inner product. Some necessary geometric notions in these spaces have been developed in Chapter 2 and will be important here.

The last three sections of the chapter provide introductions to important classes of matrices which arise frequently. They are H-contractions, dissipative matrices, and symplectic matrices.