Homework 5
Due: Apr 12th , 11:00 pm, Shanghai Local Time
Problem 1. Let B ∈ R
n×n and denote by I the identity matrix of order n.
(a) Show that if the matrix I − B is singular, then there exists a nonzero vector x ∈ R
n
such that (I − B)x = 0;
(b)Deduce that ∥B∥1 ≥ 1. (Hint: you could you the definition of matrix norm, and the fact that there exists non zero x such that||Bx||1 = ||x||1)
(c) From (b), we know if ∥A∥1 < 1, then the matrix I − A is nonsingular. Now suppose that A ∈ R
n×n with ∥A∥1 < 1. Show that
(d) Hence show that
(e) Deduce that
Problem 2. Consider the system of two simultaneous nonlinear equations in the unknowns x1 and x2, defined by
(a)What are the solutions for the simultaneous nonlinear system above?
(b)Write out the Jacobian matrix J for the solution at first quadrant.
Problem 3. From the previous problem
(a) Use the Jacobian matrix you found to determine whether the fixed point iteration will converge with proper initial guess? Please explain Why.
(b) Verify your answer using matlab and plot the error for each iteration. Please attach the code.
Problem 4. Suppose that j=1,2,3...n, for square matrix A ∈ Rn×n, in another word the largest possible value can reach is C. From last homework, we already showed for any vector x ∈ Rn, Deduce what is the 1-norm for matrix A, denoted by ||A||1?