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Eigenvalue and eigenvectors
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Eigenvalue and eigenvectors
Lessons
An eigenvector of an n×n matrix A is a non-zero vector x such that Ax=λx, for some scalar λ. The scalar λ is called the eigenvalue.
We say the eigenvector x corresponds to the eigenvalue λ.
Given an eigenvalue λ of matrix A, we can find a corresponding eigenvector x by solving
(A−λI)x=0
And finding a non-trivial solution x.
The eigenspace is the null space of the matrix A−λI. In other words, the eigenspace is a set of all solutions for the equation
(A−λI)x=0
Of course, we can find the basis for the eigenspace by finding the basis of the null space of A−λI.
We say the eigenvector x corresponds to the eigenvalue λ.
Given an eigenvalue λ of matrix A, we can find a corresponding eigenvector x by solving
The eigenspace is the null space of the matrix A−λI. In other words, the eigenspace is a set of all solutions for the equation
Of course, we can find the basis for the eigenspace by finding the basis of the null space of A−λI.
- IntroductionEigenvalues and Eigenvectors Overview:a)Definition of Eigenvalues and Eigenvectors
• What are eigenvectors?
• What are eigenvalues?b)Verifying Eigenvalues/Eigenvectors of a Matrix
• Eigenvectors: Show that Ax=λx
• Eigenvalues: Get a non-trivial solution for (A−λI)x=0
• Finding a eigenvector given an eigenvaluec)Eigenspace
• What is an eigenspace?
• Finding a basis for the eigenspace - 1.Verifying Eigenvectors
Let. Is
an eigenvector of A? If so, find the eigenvalue. What about
?
- 2.Let
. Is
an eigenvector of A? If so, find the eigenvalue.
- 3.Verifying Eigenvalues and finding a corresponding eigenvector
Let. Is λ=1 an eigenvalue of A? If so, find a corresponding eigenvector.
- 4.Finding a Basis for the Eigenspace
Find a basis for the corresponding eigenspace if: - 5.Proof Related to Eigenvalues and Eigenvectors
Prove that if A2 is the zero matrix, then the only eigenvalue of A is 0. - 6.Let λ be an eigenvalue of an invertible matrix A. Then λ−1 is an eigenvalue of A−1.