Natural and Computational Sciences Mathematics

MASSEY UNIVERSITY School of Fundamental Sciences and School of Natural and Computational Sciences Mathematics 160.102 Algebra (Auckland, Manawatū, and Distance) Assignment 2. Due 11pm Tuesday 26 May Semester One, 2021 1. Given that z = 1 + 2i is a root of p(z) = z 4 − 5z 3 + 13z 2 − 19z + 10, find all roots of p(z) and use this to express p(z) as a product of (a) 4 linear factors and (b) real linear and real irreducible quadratic factors. Check your roots in Matlab using the command roots. 2. Use the subsitution w = z 2 to find all 4 roots of z 4 + z 2 + 1 and plot them in the complex plane. 3. Use de Moivre’s Theorem (eiθ )n = einθ , equivalently (cos θ + i sin θ)n = cos(nθ) + i sin(nθ), with n = 3 to obtain the triple angle formulae that express sin(3θ) and cos(3θ) in terms of sin(θ) and cos(θ). 4. Use elementary row operations (row echelon form) to compute the determinant of   1 1 1 0 0 2 1 0   1 2 2 1 . 1 1 1 4 Check your answer in Matlab. 5. Show that for all square matrices A, if λ is an eigenvalue of A then λ2 is an eigenvalue of A2 . 1 6. Show that for all invertible square matrices A, if λ is an eigenvalue of A then 1/λ is an eigenvalue of A−1 . 7.  Whatdoes Matlab do when asked to provide the eigenvalues of an n × n matrix (such as 1 1 ) which has fewer than n linearly independent eigenvectors? How could you use 0 1 Matlab to determine when a large n × n matrix has fewer than n linearly independent eigenvectors? Illustrate your method in Matlab. 8. The Matlab command randn(n) creates an n × n matrix in which each entry is a random number chosen from the normal distribution. (a) Create such a 4 × 4 matrix and use Matlab to find its eigenvalues and eigenvectors. Use Matlab to check that the matrix of eigenvectors can be used to diagonalize the matrix. How many eigenvalues are real? If you repeat the experiment with different random numbers, do you always get the same number of real eigenvalues? (b) Create a large such matrix (without printing it out, e.g. use A = randn(100); where the semicolon stops the result being printed out) and plot its eigenvalues as points in the complex plane. What do you notice? What happens for different values of n? 9. For large matrices, computing the eigenvalues and eigenvectors the way we have been doing it by hand is prohibitively expensive, even on a computer. There is a faster method called the power method: Step 1. Choose any nonzero starting vector x0 . Step 2. Let xk+1 = Axk for k = 0, 1, 2, . . . . Step 3. Let bk = x> k xk+1 x> k xk for k = 0, 1, 2, . . . . Then the sequence b0 , b1 , b2 , . . . tends to the eigenvalue of A of largest modulus, and xk tends to  an eigenvector.    3 −1 −1 1 5  and x0 = 1. Let A = −12 0 4 −2 −1 1 In Matlab, compute b0 , b1 , b2 , b3 , and b4 . How do they compare to the largest eigenvalue of A? 2
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