Lecture 8 - 8 Eigenvalue Problems 8.1 Motivation and...

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8 Eigenvalue Problems 8.1 Motivation and Definition Matrices can be used to represent linear transformations. Their effects can be: rotation, reflection, translation, scaling, permutation, etc., and combinations thereof. These transformations can be rather complicated, and therefore we often want to decompose a transformation into a few simple actions that we can better understand. Finding singular values and associated singular vectors is one such approach. In engineering, one often speaks of principal component analysis. A more basic approach is to consider eigenvalues and eigenvectors. Definition 8.1 Let A C m × m . If for some pair ( λ, x ) , λ C , x (= 0 ) C m we have A x = λ x , then λ is called an eigenvalue and x the associated eigenvector of A . Remark Eigenvectors specify the directions in which the matrix action is simple: any vector parallel to an eigenvector is changed only in length and/or orientation by the matrix A . In practical applications, eigenvalues and eigenvectors are used to find modes of vibrations (e.g., in acoustics or mechanics), i.e., instabilities of structures can be inves- tigated via an eigenanalysis. In theoretical applications, eigenvalues often play an important role in the analysis of convergence of iterative algorithms (for solving linear systems), long-term behavior of dynamical systems, or stability of numerical solvers for differential equations. 8.2 Other Basic Facts Some other terminology that will be used includes the eigenspace E λ , i.e., the vector space of all eigenvectors corresponding to λ : E λ = span { x : A x = λ x , λ C } . Note that this vector space includes the zero vector — even though 0 is not an eigen- vector. The set of all eigenvalues of A is known as the spectrum of A , denoted by Λ( A ). The spectral radius of A is defined as ρ ( A ) = max {| λ | : λ Λ( A ) } . 8.2.1 The Characteristic Polynomial The definition of eigenpairs A x = λ x is equivalent to ( A - λI ) x = 0 . Thus, λ is an eigenvalue of A if and only if the linear system ( A - λI ) x = 0 has a nontrivial (i.e., x = 0 ) solution. 71
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This, in turn, is equivalent to det( A - λI ) = 0. Therefore we define the characteristic polynomial of A as p A ( z ) = det( zI - A ) . Then we get Theorem 8.2 λ is an eigenvalue of A if and only if p A ( λ ) = 0 . Proof See above. Remark This definition of p A ensures that the coefficient of z m is +1, i.e., p A is a monic polynomial.
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