Results 1 to 3 of 3

Math Help - Positive definite symmetric matrix has maximal elements on diagonal

  1. #1
    Newbie
    Joined
    May 2012
    From
    United States
    Posts
    17

    Positive definite symmetric matrix has maximal elements on diagonal

    Let A be a positive definite symmetric matrix. Prove that for any column (or row) the maximal element of that list is on the diagonal. I've only been able to prove the weaker statement a_(i,i)+a_(j,j)>2*a_(i,j) for i not equal to j.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member vincisonfire's Avatar
    Joined
    Oct 2008
    From
    Sainte-Flavie
    Posts
    469
    Thanks
    2
    Awards
    1

    Re: Positive definite symmetric matrix has maximal elements on diagonal

    WLOG () z^T A z = \sum\limits_{kl}z_k A_{kl} z_l = z_j A_{jj} z_j+z_i A_{ii} z_i+2 z_i A_{ij} z_j>0
    So I guess you got up to there and you set z_j = 1 and z_i = -1. You need to set z_i and z_j equal to values that are functions of A_{ij}. You should get cancellations and the desired result.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2013
    From
    United States
    Posts
    1

    Exclamation Counterexample

    I wrestled with this problem for a while, and have found a counterexample.

    Let A= \left[ \begin{array}{ccc} 1 & 0 & 0.4 \\ 0 & 1 & 0 \\ 0 & 0 & 0.4  \end{array} \right] .

    It has full rank so  x \ne 0 \Longrightarrow Ax \ne 0 \Longrightarrow x^T A^T Ax = (Ax)^T Ax = \left\Vert{Ax}\right\Vert _2 ^2 > 0 \Longrightarrow B=A^T A is symmetric positive-definite.

    B is  \left[ \begin{array}{ccc} 1 & 0 & 0.4 \\ 0 & 1 & 0 \\ 0.4 & 0 & 0.32 \end{array} \right] .

    This matrix is not diagonally dominant \left( \forall i, \vert b_{ii} \vert \ge \sum_{j \ne i}{\vert b_{ij} \vert} \right). We don't even have the weaker  \forall i, \vert b_{ii} \vert \ge \left\vert \sum_{j \ne i}{ b_{ij}} \right\vert , nor do the entries on the diagonal correspond with the 3 largest entries of the matrix. In what way can one unambiguously define "maximal entries" in which B's maximal entries lie on its diagonal?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Positive-definite Hermitian matrix
    Posted in the Algebra Forum
    Replies: 3
    Last Post: April 4th 2012, 12:41 PM
  2. positive definite matrix and inner products
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 2nd 2009, 03:49 AM
  3. 2x2 Positive Definite matrix.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 10th 2009, 05:39 PM
  4. Replies: 1
    Last Post: February 23rd 2009, 08:24 PM
  5. positive definite matrix
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 26th 2008, 10:57 AM

Search Tags


/mathhelpforum @mathhelpforum