[SOLVED] MATH307 Individual Homework7

Instructions: Read textbook pages 37 to 38 and 43 to 45 before working on the homework problems. Show all steps to get full credits.

Use Cauchy Schwartz inequality to prove the following inequalities. HInt: use the default inner product in Rⁿ and R^m×n which are defined as < u,v >=^Xu_iv_i

i

and < A,B > = ^XXa_ijb_ij respectively.

i         j

1. Prove that X|x_i| ≤ √n X|x_i|2, for all x ∈ C_n.

i                                          i

XX 2                                                      3 XX 4 XX 6

2. Prove that |a_ij| |b_ij|| ≤    |a_ij|     |b_ij| , for all A,B ∈

i         j                                                   i         j                             i         j

Cm×n.