Math 2406: Abstract Vector Spaces

Lecture 24 Plan  (Thursday, November 15, 2007).

Reminder: Test Tuesday, Nov. 20, covers 4.13-4.21, 5.1-5.13, 6.1-6.9.

Apostol 7.1-7.6.

We begin Chapter 7 on how eigenvalues relate to other properties of linear operators. We've seen that any linear transformation or matrix can be made upper triangular (in fact, a Jordan matrix) by the appropriate change of basis.  Certain linear maps can be made diagonal--they are diagonalizable. The right basis is a set of eigenvectors, an eigenbasis. Today we'll look at special class of diagonalizable maps--Hermitian operators, and see why they're diagonalizable.

Review questions:

1. What's the general form of a Hermitian or skew-Hermitian 2x2 or 3x3 matrix?
2. For the most general form of the 2x2 Hermitian and skew-Hermitian matrices, what are the orthonormal eigenbases?