5.1 - 5.1 Eigenvectors and Eigenvalues Given n n matrix A...

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5.1 Eigenvectors and Eigenvalues Given n × n matrix A The transformation vx Avx moves vectors around in R n EX: n = 2, A = b 2 3 3 2 B A b 1 0 B = b 2 3 B , A b 1 1 B = b 5 5 B , A b 1 1 B = b 1 1 B We are interested in vectors which are transformed into scalar multiples of themselves, like A b 1 1 B = 5 b 1 1 B , A b 1 1 B = 1 b 1 1 B DeFnition: A vector n = 0 such that = λvx for some λ ∈ R is called an eigenvector with eigenvalue λ . Example above: pairs of e-vector/e-values are 1 = b 1 1 B , λ 1 = 5 and 2 = b 1 1 B , λ 2 = 1 3 1 = b 3 3 B = vV 1 is another e-vector with e-value λ 1 = 5 since b 2 3 3 2 Bb 3 3 B = b 15 15 B = 5 b 3 3 B Remark: Any multiple of an e-vector is also an e-vector. 1
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Q. What vectors vx are transformed to scalar multiples of themselves ? EX: Let’s use purely geometrical arguments to Fnd eigen- vectors of some simple 2 × 2 matrices: (i) S h = b 1 1 0 1 B shear 1 (1 , 1) 1 1 1 2 y x y x −→ Only e-vectors are multiples of = b 1 0 B , since S h b 1 0 B = b 1 0 B . The e-value is λ = 1. 2
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(ii) S c = b 1 0 0 4 B scale 1 (1 , 1) 1 1 (1 , 4) y x y x −→ Multiples of vx 1 = b 1 0 B , 2 = b 0 1 B , e-values
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5.1 - 5.1 Eigenvectors and Eigenvalues Given n n matrix A...

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