Let be e, the th vector of the canonical basis of R". It is the column vector with 1 in the ith position and 0 elsewhere. For a general matrix A RX", give the result of the following Matrix product: Ae,, e A, eAe, Exercise 2. Let be a a real number. We consider the matrix A defined by: A = 0 ;). 1. Compute || A||1, ||A||2. || A||oo and the spectral radius p(A). 2. Compute A2, A and deduce a general form for A" for n E N. 3. Deduce the value of limno A". Exercise 3. Let be A = 1 2. Compute C" for every n E N. 3. Compute A" for every n N. 0 1 0 0 O 0 O 0 o O O 1 O 0 0 0 1. Show that the matrix A can be defined by block under the following form, and define the matrices B, C, DE M3 (R): A= - (B8). Me (R). Exercise 4. Let be A R*** for n E N, such that (In - A) is invertible, and P a polynom in R[X]. 1. Prove that AP(A) = P(A)A. 2. Prove that A(1n - A)- = (In - A)-A. Exercise 5. Let be A M, (R) for n E N, such that rank(A) = 1. 1. Prove that it exit u, v R"x1 such that A H 2. Prove that A trace(A) A. Exercise 6. Let be A E Rxn for n N, such that for a norm. Prove that A is invertible and that ||1|| ||(I-A)-|| < 1- || A|| We consider the Matrix A E Rxn for n E N, defined by: A = 4 1 4 01 1. Prove that A = 4(In-N), where N is to be determined. 2. Prove that ||A|| < 2 3. Give an upper bound of condo. (A). Exercise 7. Which linear system is ill conditioned. 1.001 1 1 0 1 1.001 1 ) ( ) - (8 0.9999 M (R). -1 0.0001 0.235 0.765 1) ( x ) = (0.765) Exercise 8. We consider the following linear systems: (2) Exercise 9. Let be 2 2 4.000001 (2 1) ( ) = ( 6.00 2 2 4.000001 - (3 = 3 1) ( ") = (5.99 1. Compute the condition number of the matrix of the linear systems. 2. Compute the solutions of the two linear systems, and comment. 3. Which linear system is ill conditioned. A = 6.000001 3 { ), B = ( 4 ). 2 Use the result that give the inverse of low rank perturbed matrix to evaluate B-, 5.999999 23