Solving Ax = λx

Robert M. Corless and Nicolas Fillion
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Robert M. Corless: University of Western Ontario, Applied Mathematics
Nicolas Fillion: University of Western Ontario, Applied Mathematics

Chapter Chapter 5 in A Graduate Introduction to Numerical Methods, 2013, pp 239-268 from Springer

Abstract: Abstract This chapter aims to introduce the reader to the numerical treatment of eigenvalue problems, that is, to the solution of the equation $$\mathbf{A}\mathbf{x} =\lambda \mathbf{x}$$ . This chapter is shorter than the previous one, as it relies on many notions already introduced in the context of numerical linear algebra: factoring, backward error, condition number, and residual. We examine additional factorings relevant to eigenvalue problems, namely, the Schur factoring and the Jordan canonical form. We also outline algorithms to compute eigenvalues and eigenvectors, namely, the power method and the QR algorithm. Then, a condition number of simple eigenvalues is derived and, finally, the concept of pseudospectrum is introduced to further characterize the conditioning of eigenvalue problems. ⊲

Keywords: Condition Number; Matrix Polynomial; Simple Eigenvalue; Left Eigenvector; Jordan Canonical Form (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-8453-0_5

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