3.7 Eigenvalues and Eigenvectors

Here, we will discuss briefly eigenvalues and eigenvectors of an n n matrix. We also show how they can be used to describe solutions of linear systems.

Definition 3.24

An n n matrix A is said to have an eigenvalue ? of A if there exists a nonzero vector, called an eigenvector x, such that

Then relation (3.68) represents the eigenvalue problem and we will refer to ( ?, x) as an eigenpair.

The equivalent form of (3.68) is

where I is an n n identity matrix. The system of equation (3.69) has nontrivial solutions x if, and only if, A ? ? I is singular or, equivalently,

Relation (3.70) represents a polynomial equation in ? of degree n, which in principle, could be used to obtain the eigenvalues of matrix A. This equation is called the characteristic equation of A. There are n roots of (3.70), which we will denote by ? ₁, ? ₂, , ? _n. For a given eigenvalue ? _i, the corresponding eigenvector x _i is not uniquely determined. If x is an eigenvector then so is ? x, where ? is any nonzero scalar.

Example 3.31

Find the eigenvalues and eigenvectors of the following matrix

Solution. To find the eigenvalues of the given matrix A using (3.70) , we have

which gives...