University:
University of California, Berkeley
Course:
MATH 56 Linear Algebra
Academic year:

2023
Views:

268

Pages:

2
Author:

Clint W.

→ − − x such that A → xλ Definition: An eigenvector of an n x n matrix A is a nonzero vector = for some scalar λ. Definition: A scalar λ is called an eigenvalue of A if there is a nontrivial solution of → − x − → − → − A→ = λ ; such an is called an eigenvector corresponding to λ. x x x Exercise 1. Is λ = 2 an eigenvalue of 3 3 → − x 2 ? Why or why not? (#1) 8 1 3 → − Exercise 2. Show that x = is an eigenvector of A = 1 1 eigenvalue. (Poole Section 4.1 Example 4.1) 1 and find the corresponding 3 1 2 and determine all eigenvectors corre4 3 sponding to this eigenvalue. (Poole Section 4.1 Example 4.2) Exercise 3. Show that 5 is an eigenvalue of A = Class Exercise 1. Is 1 −3 an eigenvector of 4 −3 1 ? If so, find the eigenvalue. (#3) 8    4 3 7 Class Exercise 2. Is −3 an eigenvector of −4 −5 1 2 4  9 1? If so, find the eigenvalue. (#5) 4  0 −1 3 1 ? If so, find one corresponding 4 5  3 Class Exercise 3. Is λ = 4 an eigenvalue of  2 −3 eigenvector. (#7) Exercise 4. Find a basis for the eigenspace corresponding to each listed eigenvalue. (#9) 5 0 A= , λ = 1, 5 2 1 − − x = λ→ x, Definition: The eigenspace of A corresponding to λ is the set of all solutions of A→ where λ is an eigenvalue of A. It consists of the zero vector and all eigenvectors corresponding to λ. Class Exercise 4. Find a basis for the eigenspace corresponding to each listed eigenvalue. (#11) 4 −2 A= , λ = 10 −3 9  7 Exercise 5. Show that λ = 6 is an eigenvalue of A = −3 2 eigenspace. (Poole Section 4.1 Example 4.3) Class Exercise 5. Find a basis for the eigenspace (#13, 15)   4 0 1 4 (a) A = −2 1 0, λ = 1, 2, 3 (b) A = −1 −2 0 1 2  1 −2 3 6  and find a basis for its 2 2 corresponding to each listed eigenvalue.  2 3 1 −3, λ = 3 4 9 Theorem: The eigenvalues of a triangular matrix are the entries on its main diagonal.   4 2 1 Example: The eigenvalues for the matrix  0 1 3 are 4, 1, and 7. −0 0 7 → − → − Theorem: If v 1 ,....., v r are eigenvectors that correspond to distinct eigenvalues λ1 , λ2 , . . . . . . , − − λ of an n x n matrix A, then the set { → v , ......, → v } is linearly independent. r 1 r Exercise 6. Explain why a 2 x 2 matrix can have at most two distinct eigenvalues. Explain why an n x n matrix can have at most n distinct eigenvalues. (#31) Class Exercise 6. Let λ be an eigenvalue of an invertible matrix A. Show that λ−1 is an eigenvalue of A−1 . (#33) Class Exercise 7. Show that λ is an eigenvalue of A if and only if λ is an eigenvalue of AT . (#35) Class Exercise 8. Consider an n x n matrix A with the property that the row sums all equal the same number s. Show that s is an eigenvalue of A. (#37) Homework: 2- 18 (every 4th), 22-30 even, 32, 34