Question

Let z € RM be a nonzero vector and let H= 1 - 2 VVT and P= I - WT VT where v= ||2||2e1 z 1. [Select ] Hz= = Z 2. [Select ] P2z=z 3. [Select] HPz= Pz Suggestion. All these questions can be determined algebraically, but it is tedious. Try to answer them using the geometric intuition behind the matrices defined.

select true or false

Answer 1

H = I – 2 VVT P= In VVT VTV VTV Now VTV= 2 11v112 (1) T [- 2 VVT ۱۱۷۱۱ 2 JE 2 VVT 1112 VVT VVT 4 2 VVT - I – avvT 110112 ۱۱۷۱۱ Ilvl14 I 4 VVT 4 v (vTv) VT 1114 ۱۷۱ 2 - l - 4 livi v UT + 4 VVT 11v112 11v114 - I - 4uVT +4 uvi 110112 10112 - I Hence True. So H2 = I2 = Z VVT I VVT lvl12 11v12 I tv VT VVT 1 VVT VUT 11v112 Ilvl2 ۱۱۱۱۱۹ (VTv) vi ol I 2 uut + 11vll2 11v114

I - 2VVT t VVT )۱۷۱۱ ۱۱۷۱۱2 H VVT ۱۱۷۱۱ - Р So P² - P False. So p²2 P² 2 2 P2 #2. Hence 11 STI I- 2 vui VVT 1 H.P - [ ۱۱۱2 ۱۱۱۱2 2 UNT + 2 VVT VVT 11 J VVT 11v112 Ilv))4 lvl12 11 I - 3 VVT 11u112 + av (VTV) VT 11u114 11 T- 3 VVT + 2 111 112 VVT 1/114 ۱۱2 از I - +2 VVT 3 uur 11v1 11v114 = I - VVT livl12 so H.P2= P2 - P Hence True.