[SOLVED] MATH307-Individual Homework 7

Description

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.