Eigenvalues and Eigenvectors
What are Eigenvalues and Eigenvectors?
In this chapter, we mainly focus on square matrices.
Let be an n x n matrix. We want to find a special nonzero vector in such that its transformation by is a scaling of this vector. This nonzero vector is called an eigenvector and the scaling factor is called an eigenvalue (note: it can be zero). More rigorously, we have the following definition:
Definition: An eigenvector of an n x n matrix is a nonzero vector such that for some real number , which is called an eigenvalue of . is said to be an eigenvector corresponding to .
From the geometric point of view, it means that the line containing vector i.e. remains unchanged under the linear transformation corresponding to . And any vector in will be scaled by factor under the transformation.
In the applet below, you first define the 2 x 2 matrix by setting and , you need to move the vector such that it lies on the orange dotted line containing .
Complete the following tasks:
- Find a matrix that has two eigenvalues.
- Find a matrix that has only one eigenvalue and all the eigenvectors lie on a line.
- Find a matrix that has only one eigenvalue and all nonzero vectors are eigenvectors.
- Find a matrix that has no eigenvalue.
Write down the matrices that you have obtained in the above tasks.