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# ESM3A HW2 - Jacobs University Bremen School of Engineering...

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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. Mallahi-Karai 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 0 - 1 1 2 - 1 0 0 1 0 1 1 0 1 0 , find the Jordan normal form. What is its rank, what is the basis of its null-space? 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 0 as n → ∞ ? 2.4. Solve the system of linear equations by using the LU-decomposition: 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 .

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b) Given an n × n Jordan block matrix J = λ 1 0 · · · 0 0 0 λ 1 · · · 0 0 0 0 λ · · · 0 0 . . . . . . . . . . . . . . . 0 0 0 · · · λ 1 0 0 0 · · · 0 λ . and a polynomial p ( x ) = a 0 + a 1 x + . . . + a m x m of arbitrary degree m , prove that: p ( J ) := a 0 I + a 1 J + . . . + a n J n = p ( λ ) p 0 ( λ ) 1! p 00 ( λ ) 2! · · · p ( n - 2) ( λ ) ( n - 2)! p ( n - 1) ( λ ) ( n - 1)! 0 p ( λ ) p 0 ( λ ) 1! · · · p ( n - 3) ( λ ) ( n - 3)! p ( n - 2) ( λ ) ( n - 2)! . . . . . . . . . . . . . . . 0 0 0 · · · p ( λ ) p 0 ( λ ) 1! 0 0 0 · · · 0 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 . We follow the general recipe discussed in class, i.e., to write the linear system under consid- eration ˙ x 1 ( t ) = a 11 x 1 ( t ) + a 12 x 2 ( t ) + . . . + a 1 n x n ( t ) , ˙ x 2 ( t ) = a 11 x 1 ( t ) + a 12 x 2 ( t ) + . . . + a 1 n x n ( t ) , · · · ˙ x n ( t ) = a n 1 x 1 ( t ) + a n 2 x 2 ( t ) + . . . + a nn x n ( t ) , in matrix-vector form ˙ X ( t ) = AX ( t ) , X ( t ) = x 1 ( t ) x 2 ( t ) . . . x n ( t ) , A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n .

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ESM3A HW2 - Jacobs University Bremen School of Engineering...

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