CHAPTER EIGHT Additional Topics

8.1 HERMITIAN MATRICES AND THE SPECTRAL THEOREM

It is clear that the chief end of mathematical study must be to make the student think.

—JOHN WESLEY YOUNG (1880–1932)

Cogito Ergo Sum. I think, therefore I am.

—RENÉ DESCARTES (1596–1650)

Introduction

In Section 5.3 on coordinate systems, we saw instances of a linear transformation that could be put into a simple form—at the expense of changing the basis of the underlying space. Later, in Section 5.3, we saw how to find a basis so that a given linear transformation would be represented by a diagonal matrix. This turned out to be possible in some cases, but not in all. Here we take up this theme again, and shall find that for a symmetric real matrix or Hermitian complex ...

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