Page 319 - A Course in Linear Algebra with Applications

Chapter Nine

MORE ADVANCED TOPICS

This chapter is intended to serve as an introduction to
some of the more advanced parts of linear algebra. The most
important result of the chapter is the Spectral Theorem, which
asserts that every real symmetric matrix can be diagonalized
by means of a suitable real orthogonal matrix. This result
has applications to quadratic forms, bilinear forms, conies and
quadrics, which are described in 9.2 and 9.3. The final section
gives an elementary account of the important topic of Jordan
normal form, a subject not always treated in a book such as
this.

9.1 Eigenvalues and Eigenvectors of Symmetric and
Hermitian Matrices

In this section we continue the discussion of diagonaliz-
ability of matrices, which was begun in 8.1, with special regard
to real symmetric matrices. More generally, a square complex
matrix A is called hermitian if

A = A*,

T
that is, A = (A) . Thus hermitian matrices are the complex
analogs of real symmetric matrices. It will turn out that the
eigenvalues and eigenvectors of such matrices have remark-
able properties not possessed by complex matrices in general.
The first indication of special behavior is the fact that their
eigenvalues are always real, while the eigenvectors tend to be
orthogonal.

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