Last Updated 10/28/2022

Sage Essentials
Section 7: Matrix Manipulations

Eigenvalues and Eigenvectors

Example 5

To compute the eigenvalues of a square matrix, we use the eigenvalues command. For example, let us find the eigenvalues of the matrix $A=\left[\begin{array}{cc}1& 4\\ 2& 3\end{array}\right]$ .

The eigenvalues are -1 and 5.

To find both the eigenvalues and eigenvectors, we use the eigenvectors_right command instead. Note that each eigenvector is unique up to a scalar multiple. Refer to your textbook for more information.

The results listed for each eigenvalue, one of its eigenvectors, and the multiplicity of the eigenvalue in the matrix's characteristic polynomial. For example, $\left[\begin{array}{c}1\\ 1\end{array}\right]$  is an eigenvector of the eigenvalue 5, and  $\left[\begin{array}{c}1\\ -\frac{1}{2}\end{array}\right]$  is an eigenvector of the eigenvalue -1. The characteristic polynomial of the matrix is ${\left(\lambda -\left(-1\right)\right)}^1{\left(\lambda -5\right)}^1$ .

Try this!

Find the eigenvalues of the matrix $A=\left[\begin{array}{ccc}1& 2& 1\\ 0& 3& 1\\ 0& 5& -1\end{array}\right]$ .