### Differences between the Third and Fourth Editions

The principal content change of the fourth edition is the inclusion
of a new section (Section 6.7) discussing the **singular value
decomposition** and the **pseudoinverse** of a matrix or
linear transformation between finite-dimensional inner product
spaces. Our approach is to treat this material as a generalization
of our characterization of normal and self-adjoint operators.

The organization of the fourth edition is essentially the same
as in the third edition. Nevertheless, the fourth edition contains
many significant local changes that improve the book. Section
5.1 (Eigenvalues and Eigenvectors) has been streamlined, and some
material previously in Section 5.1 has been moved to Section 2.5
(The Change of Coordinate Matrix). Further improvements include
revised proofs of some theorems, additional examples, new exercises,
and literally hundreds of minor editorial changes.

### Other Changes

- The
**List of Frequently Used Symbols** (now entitled
**List of Symbols) **has been moved to the book cover pages.
- The
**Index of Theorems** has been removed. However, a
page number is included with each reference to a theorem outside
of the section in which the theorem occurs.