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Lecture Slides 11 - AMS 210 Applied Linear Algebra AMS 210...

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AMS 210: Applied Linear Algebra October 15, 2009 AMS 210
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Topics Today Problem Set 6 Summary of Eigenvalue/vector Process Characteristic Polynomial Formulae Gaussian Elimination Gaussian Elimination Examples LU Decomposition Gauss-Jordan Elimination and Pivoting AMS 210
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Problem Set 6 This week’s problem set is due next Thursday, October 22. Read sections 3.1 and 3.2. Exercises: 3.1: 2, 3, 5, 23(i)/(ii)/(iii), 24(i)(abcd); 3.2: 2i, 3ab, 5, 6ab [that is, use the systems in subparts (a) and (b) of 3ab], 7ab [ditto], 14a [i.e. use system from 3a], 17. Show work. If multiple sheets, stapled paper is strongly preferred. AMS 210
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Summary of Eigenvector Process 1 Find and solve characteristic polynomial, i.e. det( A - λ I ) = 0 for λ . 2 For all λ found, substitute back into definition of eigenvector, i.e. Ax = λ x = λ Ix so ( A - λ I ) x = 0. λ is now a constant, so A - λ I should (unless one is proving something about general matrices) be a constant as well. 3 Solve the resulting homogenous systems, which will generally be underconstrained, with fewer equations than variables. AMS 210
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Characteristic Polynomial for 2x2 Matrix General 2x2 case: let A = a b c d . Characteristic polynomial is a - λ b c d - λ = ( a - λ )( d - λ ) - bc = λ 2 - ( a + d ) λ + ( ad - bc ) = λ 2 - Tr( A ) λ
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