1. Not finding help here? Sign up for a free 30min 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!

Symmetric matrix with eigenvalues

  1. Apr 29, 2009 #1
    1. The problem statement, all variables and given/known data
    Let {u1, u2,...,un} be an orthonormal basis for Rn and let A be a linear combination of the rank 1 matrices u1u1T, u2u2T,...,ununT. If

    A = c1u1u1T + c2u2u2T + ... + cnununT

    show that A is a symmetric matrix with eigenvalues c1, c2,..., cn and that ui is an eigenvector belonging to ci for each i.



    2. Relevant equations

    No clue...

    3. The attempt at a solution

    I'm really stuck with this problem. So I'm really just hoping for a little hint or something.
    It SOUNDS easy, but as I said, I have no clue where to start.

    Hope you can help.


    Regards
     
  2. jcsd
  3. Apr 29, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Ylle! :smile:
    (i assume you can prove that A is symmetric)

    You just need to prove that Aui = ciui

    (and remember the basis is orthonormal :wink:)
     
  4. Apr 29, 2009 #3
    Re: Eigenvalues

    Hi :)

    I'm not totally sure about the first, but I have this (See the picture added). I don't know if it is proved there ?
    For it to be symmetric it's it has to be: A = AT, right ?

    And the next I can't figure out what you mean :?
    If you add ui to A I guess you can remove the u1-n and u1-nT because they become the identitymatrix for all of them. And then you have c1-nui left. So how do I make the c1-n to ci ? :)
     

    Attached Files:

  5. Apr 30, 2009 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Ylle! :smile:

    Yes, A = AT.

    But your proof doesn't work, because transpose is like inverse … it alters the order of things … so (AB)T = BTAT :wink:
    Just write Aui in full (using the orthonormality of the u's) …

    what do you get? :smile:
     
  6. Apr 30, 2009 #5
    Re: Eigenvalues

    Well, my guess is you have:

    A = ciuiuiT

    If you then add ui on both sides you get:

    Aui = ciuiuiTui

    And because uiuiT = <ui,<ui> = ||ui|| = 1 since it's a basis, you have:

    Aui = ciui


    Don't know if that is correct ?
     
  7. Apr 30, 2009 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Ylle! :smile:

    (btw, please don't say "add" :eek: … say "multiply", or if you prefer a more neutral word, "apply" :wink:)
    Yup, that's fine …

    that's what "orthonormal" is all about! :biggrin:

    but … you still have to deal with all the u's in the ∑ that aren't that particular ui

    use the orthonormal property again. :smile:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook