Eigenvalues and Eigenvectors

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Session Overview

If the product Ax points in the same direction as the vector x, we say that x is an eigenvector of A. Eigenvalues and eigenvectors describe what happens when a matrix is multiplied by a vector. In this session we learn how to find the eigenvalues and eigenvectors of a matrix.

Session Activities

Lecture Video and Summary

• Watch the video lecture Eigenvalues and Eigenvectors (00:51:23)

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  Eigenvalues and Eigenvectors

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• Read the accompanying lecture summary (PDF)
• Lecture video transcript (PDF)

Suggested Reading

• Read Section 6.1 through 6.2 in the textbook.

Problem Solving Video

• Watch the recitation video on Eigenvalues and Eigenvectors (00:09:21)

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  Eigenvalues and Eigenvectors

  > Download from iTunes U (MP4)

  > Download from Internet Archive (MP4)

• Recitation video transcript (PDF)

Check Yourself

Problems and Solutions

Work the problems on your own and check your answers when you're done.

• Problems (PDF)
• Solutions (PDF)

Further Study

Eigenvalue Demonstrations*

These demonstrations employ Java® applets with voice-over narration by Professor Strang.

• Eigenvectors of 2 by 2 matrices
• Powers AⁿV lead toward the top eigenvalue/eigenvector

Mini-lectures on Eigenvalues

These mini-lectures with voice-over narration below help to explain some key Eigenvalue concepts.

• det(A-λI)=0
• Eigenvectors and Trace
• Powers
• Diagonalization
• Differential Equations
• Symmetry
• Positive Definite
• SVD

*Funding for these demonstrations was provided by a grant from the The d'Arbeloff Fund for Excellence in MIT Education as part of The d'Arbeloff Interactive Mathematics Project (d'A I M P).

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