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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 .

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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 .
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