# Eigenvalue and eigenvectors

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##### Intros
###### Lessons
1. Eigenvalues and Eigenvectors Overview:
2. Definition of Eigenvalues and Eigenvectors
• What are eigenvectors?
• What are eigenvalues?
3. Verifying Eigenvalues/Eigenvectors of a Matrix
• Eigenvectors: Show that $Ax=\lambda x$
• Eigenvalues: Get a non-trivial solution for $(A-\lambda I)x=0$
• Finding a eigenvector given an eigenvalue
4. Eigenspace
• What is an eigenspace?
• Finding a basis for the eigenspace
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##### Examples
###### Lessons
1. Verifying Eigenvectors
Let . Is an eigenvector of $A$? If so, find the eigenvalue. What about ?
1. Let . Is an eigenvector of $A$? If so, find the eigenvalue.
1. Verifying Eigenvalues and finding a corresponding eigenvector
Let . Is $\lambda=1$ an eigenvalue of $A$? If so, find a corresponding eigenvector.
1. Finding a Basis for the Eigenspace
Find a basis for the corresponding eigenspace if:
1. Proof Related to Eigenvalues and Eigenvectors
Prove that if $A^2$ is the zero matrix, then the only eigenvalue of $A$ is 0.
1. Let $\lambda$ be an eigenvalue of an invertible matrix $A$. Then $\lambda ^{-1}$ is an eigenvalue of $A^{-1}$.