Chapter7 - Buy from AMAZON.com...

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

View Full Document Right Arrow Icon
COPYRIGHTED It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com http://www.amazon.com/exec/obidos/ASIN/0898714540 Copyright c 2000 SIAM CHAPTER 7 Eigenvalues and Eigenvectors 7.1 ELEMENTARY PROPERTIES OF EIGENSYSTEMS Up to this point, almost everything was either motivated by or evolved from the consideration of systems of linear algebraic equations. But we have come to a turning point, and from now on the emphasis will be different. Rather than being concerned with systems of algebraic equations, many topics will be motivated or driven by applications involving systems of linear differential equations and their discrete counterparts, difference equations. For example, consider the problem of solving the system of two first-order linear differential equations, du 1 /dt = 7 u 1 4 u 2 and du 2 /dt = 5 u 1 2 u 2 . In matrix notation, this system is u 1 u 2 = 7 4 5 2 u 1 u 2 or, equivalently, u = Au , (7 . 1 . 1) where u = u 1 u 2 , A = 7 4 5 2 , and u = u 1 u 2 . Because solutions of a single equation u = λu have the form u = α e λt , we are motivated to seek solutions of (7.1.1) that also have the form u 1 = α 1 e λt and u 2 = α 2 e λt . (7 . 1 . 2) Differentiating these two expressions and substituting the results in (7.1.1) yields α 1 λ e λt = 7 α 1 e λt 4 α 2 e λt α 2 λ e λt = 5 α 1 e λt 2 α 2 e λt α 1 λ = 7 α 1 4 α 2 α 2 λ = 5 α 1 2 α 2 7 4 5 2 α 1 α 2 = λ α 1 α 2 .
Image of page 1

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

View Full Document Right Arrow Icon
COPYRIGHTED It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com http://www.amazon.com/exec/obidos/ASIN/0898714540 Copyright c 2000 SIAM 490 Chapter 7 Eigenvalues and Eigenvectors In other words, solutions of (7.1.1) having the form (7.1.2) can be constructed provided solutions for λ and x = α 1 α 2 in the matrix equation Ax = λ x can be found. Clearly, x = 0 trivially satisfies Ax = λ x , but x = 0 provides no useful information concerning the solution of (7.1.1). What we really need are scalars λ and nonzero vectors x that satisfy Ax = λ x . Writing Ax = λ x as ( A λ I ) x = 0 shows that the vectors of interest are the nonzero vectors in N ( A λ I ) . But N ( A λ I ) contains nonzero vectors if and only if A λ I is singular. Therefore, the scalars of interest are precisely the values of λ that make A λ I singular or, equivalently, the λ ’s for which det( A λ I ) = 0 . These observations motivate the definition of eigenvalues and eigenvectors. 66 Eigenvalues and Eigenvectors For an n × n matrix A , scalars λ and vectors x n × 1 = 0 satisfying Ax = λ x are called eigenvalues and eigenvectors of A , respectively, and any such pair, ( λ, x ) , is called an eigenpair for A . The set of distinct eigenvalues, denoted by σ ( A ) , is called the spectrum of A .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern