1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inner Product Question

  1. Jul 28, 2017 #1
    1. The problem statement, all variables and given/known data
    upload_2017-7-28_18-15-22.png

    2. Relevant equations
    upload_2017-7-28_18-15-30.png

    3. The attempt at a solution

    I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

    Follow all the procedures I already did:

    upload_2017-7-28_18-18-29.png

    upload_2017-7-28_18-18-46.png
    upload_2017-7-28_18-19-33.png
     

    Attached Files:

  2. jcsd
  3. Jul 28, 2017 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Hint: if ##{\bf x} = (x_1,0)##, it is easy to show that ##\langle {\bf x,x} \rangle > 0## for any ##x_1 \neq 0##. Also: for ##x_2 \neq 0## we have ##(x_1,x_2) = x_2(x_1/x_2,1)##, so ##\langle {\bf x,x} \rangle = x_2^2 \langle {\bf u,u} \rangle##, where ##{\bf u} = (x_1/x_2,1) \equiv (t,1)##.
     
  4. Jul 29, 2017 #3

    ehild

    User Avatar
    Homework Helper
    Gold Member

  5. Jul 29, 2017 #4

    StoneTemplePython

    User Avatar
    Gold Member

    are you allowed to use spectral theory here?

    Your matrix

    ##
    \begin{bmatrix}
    a & b\\
    b & c
    \end{bmatrix}##

    is real symmetric. If the determinant is positive (one of your conditions in the iff) and the trace is positive (implied by ##a \gt 0##... why?) then it this tells you...
     
  6. Jul 31, 2017 #5
    upload_2017-7-31_21-22-7.png

    upload_2017-7-31_21-22-39.png

    upload_2017-7-31_21-23-18.png
     

    Attached Files:

  7. Jul 31, 2017 #6

    ehild

    User Avatar
    Homework Helper
    Gold Member

    If you factorize Δ you get Δ=4x22(b2-4ac)
     
  8. Aug 1, 2017 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Why do you suppose I wrote ##\langle {\bf x,x} \rangle = x_2^2 \langle {\bf u,u} \rangle##, where ##{\bf u} = (t,1)?## The fact that ##t = x_1/x_2## does not really matter at all if all you want to know is the sign of ##\langle {\bf u,u} \rangle##---think about it!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Inner Product Question
Loading...