MATH 456 Homework 1Due 1/29/2023 by 11:59pm (upload to Canvas)Submission Instructions:Your submission should contain two total files. One pdf file containing all of your solutions...

Finish it asap, and please use Matlab to solve question 5 and 6


MATH 456 Homework 1 Due 1/29/2023 by 11:59pm (upload to Canvas) Submission Instructions: Your submission should contain two total files. One pdf file containing all of your solutions (including the output of your code) and one file with your code. I should not need to run your code to view your solutions. 1. (10 pts) Prove that a square matrix and its transpose have the same characteristic polynomial, and therefore the same set of eigenvalues. 2. (15 pts) An eigenvalue and eigenvector of the matrix A may be evaluated by solving the system of nonlinear equations (A− λI)x = 0 xTx = 1 for the unknowns λ and x. Using Newton’s method, starting with estimates λ0 and x0, show that the next iteration is determined by A∆x−∆λx0 − λ0∆x = −(A− λ0I)x0 −xT0 ∆x = 1 2 (xT0 x0 − 1) where ∆x = x1 − x0 and ∆λ = λ1 − λ0. Comment on the difference between this method and the method of inverse iteration. Hint: For a vector valued function f(z) with z = Rn, Newton’s method has the form zn+1 = zn − Jf(zn)−1f(zn) where Jf(zn) is the Jacobian of f evaluated at zn. Note, you will not need to compute the inverse to derive the above expression. 3. (10 pts) Assume that A is a 3× 3 matrix with the given eigenvalues. Decide to which eigenvalue Power Iteration will converge and determine the convergence rate constant S. (a) {1, 2, 7} (b) {8,−9, 10} 4. (10 pts) Carry out two steps of inverse iteration for the matrix A = ( 2 2 2 5 ) using the eigenvalue estimate λ̃ = 5 and the initial vector v0 = (1, 1) T . Verify that the elements of the vector v2 agree wth those of the true eigenvector with an accuracy of about 5%. Evaluate the Rayleigh quotient using the vector v2, and verify that the result agrees with the true eigenvalue to about 1 in 3000. 1 Computer Problems (You may use Matlab or Python) Submit your code and a table of iterates containing your results. 5. (15 pts) Let A =  5 2 −2−12 −19 12 −12 −22 15  (a) Apply 10 steps of the Power Method with initial vectors v = (1, 1, 1)T to estimate the dominant eigenvalue of A. (b) Apply 10 steps of the Inverse Power Method with shift 0 and initial vector v = (1, 1, 1)T to estimate the eigenvalue closest to zero. Provide a table of iterates for each method’s approximation to the eigenvalue. 6. (20 pts) Consider the symmetric matrix A = 2 1 11 3 1 1 1 4  . With initial guess v0 = ( 1√ 3 , 1√ 3 , 1√ 3 )T , apply the following methods: (a) Rayleigh quotient iteration (b) Inverse power iteration with a shift of 5. Provide a table of iterates for each method’s approximation to the eigenvalue and comment on the speed at which each method converges. 2