Week03_Lec04_Eigenvalues_Eigenvector_annotate - Eigenvalues and Eigenvectors In this chapter we will cover one fundamental property of matrices that

Week03_Lec04_Eigenvalues_Eigenvector_annotate -...

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Eigenvalues and Eigenvectors In this chapter, we will cover one fundamental property of matrices, that is eigenvectors and eigenvalues. They have many applications in engineering and science, for example, advanced dynamics, vibration analysis (natural frequency of the bridge and the design of car stereo system), control theory, electric circuits and quantum mechanics. Furthermore, the Google algorithm calculates an eigenvector of the multi-billion entry matrix that is the entire world-wide web! Learning Outcomes On successful completion of this chapter, you should be able to: (i) understand what eigenvalues and eigenvectors are, (ii) find the eigenvalues and eigenvectors for n × n matrices. 1
Given a matrix R, almost all vectors change in direction when multiply with R. Example: Given a vector w = [3, 1] T and matrix R = 0 1 2 1 , we have R w = 0 1 2 1 3 1 = 1 5 However, there may exist some vectors v , such that, when multiply with R, the new vector R v is parallel to v and the magnitude of R v is scaled by a factor λ (any real number). Note: if λ is positive, then R v is in the same direction as v .

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