Differential Equations and Linear Algebra, 4th Edition

6 Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues

Given a square matrix A, let us pose the following question: Does there exist a nonzero vector v such that the result Av of multiplying v by the matrix A is simply a scalar multiple of v? Thus we ask whether or not there exist a nonzero vector v and a scalar $\lambda$ such that

$$\mathbf{\text{Av}}=\lambda \mathbf{\text{v}}.$$ (1)

Section 6.3 and subsequent chapters include interesting applications in which this question arises. The following definition provides the terminology that we use in discussing Eq. (1).

Definition Eigenvalues and Eigenvectors

The number $\lambda$ is said to be an eigenvalue of the $n\times n$ matrix A provided there exists a nonzero vector v such that

$$\mathbf{\text{Av}}=\lambda \mathbf{\text{v}},$$ (1)

in which case the vector v is called an eigenvector of the ...