Orthogonal and Unitary Transformations

Abstract

In Chapters 7 and 8, general theorems were proved about the structure of a single linear transformation. For certain types of linear transformations, and matrices corresponding to them, over the fields of real or complex numbers, sharper theorems about the eigenvalues and eigen-vectors can be proved. This chapter contains some of these results, for orthogonal and symmetric transformations on vector spaces over the real numbers, with applications to quadratic forms, and for unitary, self-adjoint, and normal transformations on vector spaces over the complex numbers. Further results, about the exponential of a matrix, and the Perron-Frobenius theorem on the eigenvalues of positive real matrices, with applications to systems of differential equations and Markov chains, are proved using analytic methods in Sections 34 and 35.