Linear Algebra, 5th Edition

5.1 Eigenvalues and Eigenvectors

In Example 3 of Section 2.5, we were able to obtain a formula for the reflection of ${\text{R}}^2$ about the line $y=2x$. The key to our success was to find a basis ${\beta }^{\prime }$ for which ${[\text{T}]}_{{\beta }^{\prime }}$ is a diagonal matrix. We now introduce the name for an operator or matrix that has such a basis.

Definitions.

A linear operator T on a finite-dimensional vector space V is called diagonalizable if there is an ordered basis $\beta$ for V such that ${[\text{T}]}_{\beta }$ is a diagonal matrix. A square matrix A is called diagonalizable if ${\text{L}}_A$ is diagonalizable.

We want to determine when a linear operator T on a finite-dimensional vector space V is diagonalizable and, if so, how to obtain an ordered basis $\beta =\{v_1,v_2,\dots ,v_n\}$ for V such that ${[\text{T}]}_{\beta }$ is a diagonal matrix. Note ...