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4.7 Change of Basis 241 4.8 Applications to Difference Equations 246 4.9 Applications to Markov Chains 255 Supplementary Exercises 264 Chapter 5 Eigenvalues and Eigenvectors 267 INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls 267 5.1 Eigenvectors and Eigenvalues 268 5.2 The Characteristic Equation 276 5.3 Diagonalization 283 5.4 Eigenvectors and Linear Transformations 290 5.5 Complex Eigenvalues 297