Results 1 to 2 of 2

Thread: inner product

  1. #1
    Sep 2008

    Wink inner product

    Please, help!
    Let V be a finite dimensional inner product space over C. Suppose that T is a positive operator on V ( called positive semidefinite by some authors ). Prove that T is invertible if and only if < Tv,v > > 0 for every v is in V \ { 0 }.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Aug 2007
    Leeds, UK
    How much do you know about positive operators? If you know that they are diagonalisable then this result follows fairly easily: There is an orthonormal basis of V with respect to which T has a diagonal matrix D, whose diagonal entries are the eigenvalues of T. The matrix D has an inverse if and only if none of these eigenvalues is 0. That is equivalent to the condition $\displaystyle \langle Tv,v\rangle > 0$ for all v≠0.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Oct 18th 2011, 04:40 AM
  2. Replies: 6
    Last Post: Sep 7th 2010, 09:03 PM
  3. multivariable differential for inner product(scalar product)?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Oct 23rd 2009, 05:40 PM
  4. Replies: 4
    Last Post: Sep 2nd 2009, 04:07 AM
  5. Replies: 1
    Last Post: May 14th 2008, 11:31 AM

Search Tags

/mathhelpforum @mathhelpforum