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!

What are the matrices B, C, D?

  1. Aug 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Let x = (x1,x2) [itex]\in[/itex] Rn, x1 [itex]\in[/itex] Rn1, x2 [itex]\in[/itex] Rn2, n1 + n2 = n and A [itex]\in[/itex] Rnxn be symmetric and positive definite.
    a) Let x0 [itex]\in[/itex] Rn. Show that we can write (x-x0)TA(x-x0) = ||L(x-x0||22. Is L unique?
    b) Consider the quadratic term b = xTAx. Show that we can write b = x1TBx1 + 2x1TCx2 + x2T
    Dx2, and what are the matrices B, C, D?
    2. Relevant equations



    3. The attempt at a solution
    a) Is it using the definition of symmetric & positive defintie matrices.
    b) Isn;t that just a quadratic form?
     
  2. jcsd
  3. Aug 11, 2011 #2
    Re: Quadratic

    For (a), perhaps you should look into the square root of a matrix and under what conditions it is unique.

    For (b), you can solve this by partitioning A into appropriately sized blocks and carrying out block multiplication. And remember that A is symmetric! You'll need that fact to finish the last step.
     
  4. Aug 11, 2011 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Re: Quadratic

    Hint for (a): Look at *Cholesky Factorization* (Google search).

    RGV
     
  5. Aug 14, 2011 #4
    Re: Quadratic

    Never done Cholesky factorization before in my life.
     
  6. Aug 14, 2011 #5
    Re: Quadratic

    Do you mean the square root of positive definite symmetric matrix?
    Not sure what you mean for b)?
     
  7. Aug 14, 2011 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Re: Quadratic

    So what? I gave a suggestion and it is up to you to take the advice or not.

    RGV
     
  8. Aug 15, 2011 #7
    Re: Quadratic

    A little stuck on b), what do you partitioning A into appropriately sized blocks and carry out block multiplication.
     
  9. Aug 15, 2011 #8
    Re: Quadratic

    You can split matrices into blocks (they must be the appropriate sizes so that the multiplication is defined) and multiply them. It's quite helpful in some proofs and helps with notational issues.

    Look http://en.wikipedia.org/wiki/Block_matrix#Block_matrix_multiplication".

    And yes, look into the square root of a positive semidefinite matrix. BTW, it's related to the factorization that Ray Vickson mentioned. Depending on where you read up on this, you might see that factorization and square roots mentioned in the same chapter/article/section, etc.
     
    Last edited by a moderator: Apr 26, 2017
  10. Aug 17, 2011 #9
    Re: Quadratic

    b = xTAx = (x1 x2)TA(x1 x2)
    where A has entries B = a11 C = a12 = a21 (symmetric matrix) and D = a22
    b = x1TBx1 + x1TCx2 + x2TCx1 + x2TDx2
    = x1TBx1 + 2x1TCx2 + x2TDx2

    But how do we define B, C, D?
    Is it just similar to the wiki page?
     
    Last edited: Aug 17, 2011
  11. Aug 17, 2011 #10
    Re: Quadratic

    Any ideas?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook