{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

c5s1 - 5.1 Eigenvalues and Eigenvectors 1 Chapter 5...

This preview shows pages 1–3. Sign up to view the full content.

5.1 Eigenvalues and Eigenvectors 1 Chapter 5. Eigenvalues and Eigenvectors 5.1 Eigenvalues and Eigenvectors Definition 5.1. Let A be an n × n matrix. A scalar λ is an eigenvalue of A if there is a nonzero column vector v R n such that Av = λv . The vector v is then an eigenvector of A corresponding to λ . Note. If Av = λv then Av λv = 0 and so ( A λ I ) v = 0. This equation has a nontrivial solution only when det( A λ I ) = 0. Definition. det( A λ I ) is a polynomial of degree n (where A is n × n ) called the characteristic polynomial of A , denoted p ( λ ), and the equation p ( λ ) = 0 is called the characteristic polynomial . Example. Page 300 number 8.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
5.1 Eigenvalues and Eigenvectors 2 Theorem 5.1. Properties of Eigenvalues and Eigenvectors. Let A be an n × n matrix. 1. If λ is an eigenvalue of A with v as a corresponding eigenvector, then λ k is an eigenvalue of A k , again with v as a corresponding eigenvector, for any positive integer k . 2. If λ is an eigenvalue of an invertible matrix A with v as a corresponding eigenvector, then λ = 0 and 1 is an eigenvalue of A 1 , again with v as a corresponding eigenvector. 3. If λ is an eigenvalue of
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 3

c5s1 - 5.1 Eigenvalues and Eigenvectors 1 Chapter 5...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online