How to prove that the inner product is positive unless $Ax = b$?

Question

enter image description here

Suppose $Ax =b$, then the equation above = $0$

Spp $Ax \neq b$, since $A$ is positive definite, then enter image description here

Am I going to the right direction for this proof? How can I show the rest is positive as well?

score 1 · Answer 1 · 2015-04-26T12:20:45.760

1

Hint: $r=Ae$, so $\langle r,e\rangle = \langle Ae,e\rangle$.

edited Apr 26 '15 at 12:20

answered Apr 26 '15 at 12:17

I know that r = Ae, but I don't know how to use that . Can you explain more please? – shimura Apr 26 '15 at 12:19
Added a bit more. – Apr 26 '15 at 12:21
yes, I got <r,e>=e*Ae, if I expand more, it goes back to the same equation I had :( – shimura Apr 26 '15 at 12:28
$A$ is positive definite. What can you say about $\langle Ae,e\rangle$? – Apr 26 '15 at 12:28
Then its positive. Thank you so much . I didn't look at that way, I was thinking it has to be x*Ax – shimura Apr 26 '15 at 12:31

1 Answers1