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A certian Linear Algebra gimmick needed for a part of my project

  1. Jan 14, 2014 #1
    1. I need to prove that for any matrix A(n,n) and a vector v(n,1) the following is true...

    vTAv=vTATv


    So far I wasn't able to think of anyway for proving this... any help will be appreciated.
     
  2. jcsd
  3. Jan 14, 2014 #2
    I do understand intuitively how this can be true...
    Because of the arrangement of the equation, both side of the equation gets the same value.
    I am able to show that with small example,,

    [tex]
    A=\left[\begin{array}{cc}
    p & q\\
    r & s
    \end{array}\right],v=\left[\begin{array}{cc}
    a & b\end{array}\right]

    \Rightarrow v'Av=v'A'v=a^{2}p+abr+abq+b^{2}s
    [/tex]

    But I need rigorous proof for the same...

    Thank you. :)
     
  4. Jan 14, 2014 #3

    ehild

    User Avatar
    Homework Helper
    Gold Member

    What is the dimension of vTAv and vTATv?
    What is (vTAv)T? How do you transpose a matrix product?



    ehild
     
  5. Jan 14, 2014 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    For two matrices A and B, ##\left(AB\right)^T=B^TA^T##. If you don't already know that prove it using sigma notation.
     
  6. Jan 14, 2014 #5
    So this is what I understand...

    since the equation [tex]v'Av[/tex] is a number... we can write...

    [tex]v'Av=(v'Av)'=v'(v'A)'=v'A'v[/tex]

    Am I correct?
     
  7. Jan 14, 2014 #6

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Yes, that was I wanted you to do.

    ehild
     
  8. Jan 14, 2014 #7

    Mark44

    Staff: Mentor

    aashish.v, when you post a problem, include your work in the inital post. Without the work you show in your 2nd post, your 1st post would normally earn you a warning for an unacceptable homework request.
     
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