Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Positive definite matrix

  1. Feb 20, 2010 #1
    I am curious to know how g'Bg compares with g'g when B is a positive definite matrix and g is a vector.

    Is g'Bg >= g'g ?


    Thanks,
    Karthik
     
  2. jcsd
  3. Feb 20, 2010 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Well, let's look at the simplest case.

    No wait, 0x0 matrices are too simple. Let's try the next simplest case -- 1x1 matrices.

    Try analyzing the 1x1 case. What do you see?
     
  4. Feb 20, 2010 #3
    The relation seems to hold true in this case. Say, if B is 3, then g' *3*g is definitely greater than g'g
     
  5. Feb 20, 2010 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Well, there are more 1x1 matrices than just [3]! You should try a few.

    Anyways, rather than looking at 1x1 matrices one at a time, you should try proving it for all 1x1 matrices at once! You'll need to introduce one or more additional variables, of course.
     
  6. Feb 20, 2010 #5
    Yeah.. Understood. Eigen values of B has to be greater than 1 for the relation to hold true.

    Thanks
     
  7. Feb 20, 2010 #6

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I believe that statement.
     
  8. Feb 22, 2010 #7

    jvc

    User Avatar

    I think it depends on the eigenvalues of B
    if the eigenvalues of B is larger than 1, the statement holds. Otherwise, it does not.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Positive definite matrix
Loading...