Homework4solutions

Homework4solutions - Homework 4 solutions: 1. Av = Iv...

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Homework 4 solutions: 1. Av Iv λ = G G Condition for eigenvalues are det( ) 0 AI = cos sin det 0 sin cos i i e e φ θλ θ θθ ⎛⎞ = ⎜⎟ −− ⎝⎠ From which 1 = ± to find eigenvalues 11 22 1 vv A From which we get two equations, but we need only one equation since those two equations are linearly dependent because of the condition det( ) 0 = . Case 1: 1 = 12 cos sin i ve v 1 v + = Set = 1 since the above equation has infinite set of solutions and solve for . 1 v 2 v 2 1c o s sin i = After multiplying through by sin i e the normalized eigenvector becomes sin 1 o s c o s i e v = G corresponding to 1 = Case 2: 1 =− cos sin i v 1 v + Proceeding as before, then we have
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sin 1 1c o s 22 c o s i e v φ θ ⎛⎞ = + + ⎝⎠ G corresponding to 1 λ = − 2. Before proceeding it is important to make sure that vv G G =1 for both eigenvectors. The columns of the matrix V are then the corresponding eigenvectors and the diagonal elements of the matrix B are the corresponding eigenvalues corresponding to the
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Homework4solutions - Homework 4 solutions: 1. Av = Iv...

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