Question

Hi there! I'm looking for someone who can help me understanding the complex inner product. I've come up with a proof that doesn't involve any conjugates and results in a different outcome: http://codepad.org/Zp6e7WoN. The problem is that I can't find the flaw in my reasoning. To me the proof is 100% true. Is there anyone who can help me with this please?

Answer 1

U and V are complex vectors. _n_a will denote the real part and _n_b will denote the complex part.
I'm assuming the law of cosine holds up in complex space:
||U||^2 + ||V||^2 - 2*||U||*||V||*cos(angle) = ||U-V||^2

||U||^2 = U_1_a^2 + U_1_b^2 + ... + U_n_a^2 + U_n_b^2
||V||^2 = V_1_a^2 + V_1_b^2 + ... + V_n_a^2 + V_n_b^2

||U-V||^2 = (U_1_a-V_1_a)^2 + (U_1_b-V_1_b)^2 + ... + (U_n_a-V_n_a)^2 + (U_n_b-V_n_b)^2
||U-V||^2 = ||U||^2 + ||V||^2 - 2*(U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

||U||^2 + ||V||^2 - 2*||U||*||V||*cos(angle) = ||U-V||^2
=
||U||^2 + ||V||^2 - 2*(U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

subtracting ||U||^2 + ||V||^2 from both sides gives:
-2*||U||*||V||*cos(angle) = -2*(U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

dividing both sides by -2 gives:
||U||*||V||*cos(angle) = (U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

so the complex inner product should be defined by (U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b) which doesn't involve any conjugates or the presence of i.

Answer 2

the url says 404 not found post or screen shot it maybe it will be useful ill help

Answer 3

the url is the same as my second comment

Answer 4

when i go to the link it still works

Answer 5

@ozzy22 omit the period :)

Answer 6

was this your Q
||U||^2 + ||V||^2 - 2*||U||*||V||*cos(angle) = ||U-V||^2

||U||^2 = U_1_a^2 + U_1_b^2 + ... + U_n_a^2 + U_n_b^2
||V||^2 = V_1_a^2 + V_1_b^2 + ... + V_n_a^2 + V_n_b^2

||U-V||^2 = (U_1_a-V_1_a)^2 + (U_1_b-V_1_b)^2 + ... + (U_n_a-V_n_a)^2 + (U_n_b-V_n_b)^2
||U-V||^2 = ||U||^2 + ||V||^2 - 2*(U_1_a*V_1_a + U_1_b*V_1_b + ... + U_n_a*V_n_a + U_n_b*V_n_b)

||U||^2 + ||V||^2 - 2*||U||*||V||*cos(angle) = ||U-V||^2
=

Answer 7

yeah

Answer 8

another link: http://justpaste.it/hnvr

Answer 9

can you see what I'm doing wrong?

Answer 10

sorry let @SithsAndGiggles help u on this i'm confused.

Answer 11

ok

Answer 12

are you reading this SithsAndGiggles?

Answer 13

this is what i got for u

Answer 14

url

Answer 15

https://answers.yahoo.com/question/index?qid=20081209170452AAZSLpH

Answer 16

this is the answer for your problem

Answer 17

hope it answer your Q if did pls fan and medal