Linear Algebra, 5th Edition

5 Diagonalization

1. 5.1 Eigenvalues and Eigenvectors

2. 5.2 Diagonalizability

3. 5.3* Matrix Limits and Markov Chains

4. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem

This chapter is concerned with the so-called diagonalization problem. For a given linear operator T on a finite-dimensional vector space V, we seek answers to the following questions.

1. Does there exist an ordered basis $\beta$ for V such that ${[\text{T}]}_{\beta }$ is a diagonal matrix?

2. If such a basis exists, how can it be found?

Since computations involving diagonal matrices are simple, an affirmative answer to question 1 leads us to a clearer understanding of how the operator T acts on V, and an answer to question 2 enables us to obtain easy solutions to many practical problems that can be formulated ...