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Unformatted text preview: Jacobs University Bremen School of Engineering and Science Peter Oswald Fall Term 2010 120201 ESM3A — Problem Set 2 Issued: 15.9.2010 Due: Wednesday 22.9.2010 (in class) This homework deals with the Jordan normal form, its applications, and the LU factorization. Consult the textbook by G. Strang [S] (or any other text on the subject of matrix calculations and linear algebra) and the script [MK] by K. MallahiKarai for inspiration. For your convenience, the short (but correct!) version of the two application examples (linear differential equations, linear recursions) discussed in class is appended. 2.1. For the matrix A = 2 1 1 1 2 1 0 1 1 1 0 1 , find the Jordan normal form. What is its rank, what is the basis of its nullspace? 2.2. Solve the homogeneous system of linear differential equations ˙ x 1 ( t ) = 3 x 1 ( t ) + 2 x 2 ( t ) ˙ x 2 ( t ) = 4 x 1 ( t ) x 2 ( t ) using diagonalization resp. Jordan normal form. 2.3. a) Find the general solution of the linear recursion x n +1 = 2 x n + 5 x n 1 + x n 2 8 , n ≥ 3 , i.e., by a formula for x n involving arbitrary starting values x 1 ,x 2 ,x 3 or other 3 constants of your choice. b) Suppose x 1 = 1 = x 2 = 1, is it possible to choose a real number x 3 such that x n → as n → ∞ ? 2.4. Solve the system of linear equations by using the LUdecomposition: 4 x + 6 y + 9 z = 1 16 x + 25 y + 37 z = 2 28 x + 57 y + 25 z = 4 2.5. a) Find the powers of the n × n matrix H H = 0 1 0 ··· 0 0 0 0 1 ··· 0 0 0 0 0 ··· 0 0 . . . . . . . . . . . . . . . 0 0 0 ··· 0 1 0 0 0 ··· 0 0 . b) Given an n × n Jordan block matrix J = λ 1 0 ··· 0 0 λ 1 ··· 0 0 0 0 λ ··· 0 0 . . . . . . . . . . . . . . . 0 0 0 ··· λ 1 0 0 0 ··· λ . and a polynomial p ( x ) = a + a 1 x + ... + a m x m of arbitrary degree m , prove that: p ( J ) := a I + a 1 J + ... + a n J n = p ( λ ) p ( λ ) 1! p 00 ( λ ) 2! ··· p ( n 2) ( λ ) ( n 2)! p ( n 1) ( λ ) ( n 1)! p ( λ ) p ( λ ) 1! ··· p ( n 3) ( λ ) ( n 3)! p ( n 2) ( λ ) ( n 2)! . . . . . . . . . . . . . . . ··· p ( λ ) p ( λ ) 1! ··· p ( λ ) . 2.6. Using the result of the previous problem (and what you have learned in calculus classes), find A 50 , e A , and sin( A ) for the matrix A = 3 1 1 1 . Note: You are allowed to use the result of Problem 2.5 b) even if you haven’t proved it! The two examples from class Example 1 . Application of the Jordan normal form to computing the general solution of a homogeneous system of linear differential equations with constant coefficients ....
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This note was uploaded on 02/06/2011 for the course ESM 3A taught by Professor Prof. dr. peter oswald during the Spring '11 term at Jacobs University Bremen.
 Spring '11
 Prof. Dr. Peter Oswald

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