In this lesson on diagonalizing a matrix and powers of A, students learn how to find eigenvalues and eigenvectors, and then how to use them to diagonalize a matrix. The diagonalization process involves putting the eigenvectors in the columns of a matrix S, and then using S inverse A S to create a diagonal matrix with the eigenvalues as the diagonal entries. The lesson also discusses the relationship between the powers of a matrix and its eigenvalues and eigenvectors. Students learn that if all eigenvalues are different, the eigenvectors are automatically independent, but if some eigenvalues are repeated, further investigation is needed to determine if there are independent eigenvectors.
Diagonalizing a matrix and Powers of A -- Lecture 22. Learn how to diagonalize a matrix and find Eigenvalues for powers of a matrix A.
Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 23, 2008). License: Creative Commons BY-NC-SA.
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Powers A^n Demo
- Demonstration of finding Powers A^nV, which lead toward the top eigenvalue / eigenvector.
Power Method Applet
- Power Method Java applet, which was used in the demo, for you to play around with.
This lecture begins to explain why Eigenvalues are useful. You will learn how to diagonalize a matrix using Eigenvalues and Eigenvectors, and how to use them by taking a matrix A to a power. An interesting and powerful video.