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Unformatted text preview: 5.1 Eigenvectors and Eigenvalues Given n × n matrix A The transformation x → Ax moves vectors around in Rn EX: n = 2, 1 0 A= 2 3 23 32 1 1 = 5 5 ,A −1 1 = 1 −1 A = ,A We are interested in vectors which are transformed into scalar multiples of themselves, like A 1 1 =5 1 1 ,A −1 1 = −1 −1 1 Definition: A vector x = 0 such that Ax = λx for some λ ∈ R is called an eigenvector with eigenvalue λ. Example above: pairs of e-vector/e-values are x1 = 1 1 3 3 23 32 , λ1 = 5 and x2 = −1 1 , λ2 = −1 3x1 = since = v1 is another e-vector with e-value λ1 = 5 3 3 = 15 15 =5 3 3 Remark: Any multiple of an e-vector is also an e-vector. 1 Q. What vectors x are transformed to scalar multiples of themselves ? EX: Let’s use purely geometrical arguments to find eigenvectors of some simple 2 × 2 matrices: (i) Sh = 11 01 shear y y 1 (1, 1) −→ 1 1 x 1 x 2 Only e-vectors a...
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This note was uploaded on 04/09/2011 for the course MATH 232 taught by Professor Russel during the Spring '10 term at Simon Fraser.

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