s11week10 - Math 205 Spring 2011 Homework 10 due April 4 Chapter 5 Section 8 and Chapter 6 Sections 1-2 2 Week 10 5.8 Diagonalization(Mon 6.1 6.2 Second

# s11week10 - Math 205 Spring 2011 Homework 10 due April 4...

• Notes
• 17

This preview shows page 1 - 5 out of 17 pages.

Math 205, Spring 2011 Homework 10: due April 4 Chapter 5, Section 8 and Chapter 6, Sections 1-2.
2 Week 10: 5.8 Diagonalization (Mon) 6.1, 6.2 Second Order Linear DE (Fri) [reprint, week09: Eigenvalues and Eigenvectors] + diagonaliization 1.5.8 Eigenspaces, Diagonalization—————A vectorvectorvnegationslash=vector0 inRn(or inCn) is aneigenvector with eigenvalueλof ann-by-nmatrixAifAvectorv=λvectorv.We re-write the vector equation as (AλIn)vectorv=vector0,which is a homogeneous system with coef matrix (AλI),andwe wantλso that the system has a non-trivial solution.We see that the eigenvalues are the roots of thecharacteristic polynomialP(λ) = 0,whereP(λ) = det(AλI).To find the eigenvectors we find the distinctrootsλ=λi,and for eachisolve (AλiI)vectorv=vector0.—————Problem 1.Find the eigenvalues and eigenvectorsofA=parenleftbigg3151parenrightbigg
Problem 2.Find the eigenvalues and eigenvectorsofA=101280208126.
4