6.1 Eigenvalues of matrices

Although we have formally defined eigenvalues and eigenvectors for linear transformations, we often talk about them in context of matrices. (Because, as we have seen, matrices and linear transformations are two faces of the same coin.) Let’s start by translating the definition into the language of matrices.

If A ∈ℝ^n×n is a matrix, Definition 23 translates to the following: the scalar λ and the vector x ∈ℝ ∖{0} is an eigenvalue-eigenvector pair of the matrix if

Ax = λx

(6.2)

holds. This can be simplified: as the linear transformation x→λx corresponds to the matrix λI, (6.2) is equivalent to

(A − λI)x = 0

(6.3)

If you recall Chapter 4, Section 4.1.1, where we learned how matrices arise from linear transformations, you might ask the question: won’t the eigenvalues depend on the choice of the matrix?

The following theorem...