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!

Linear Algebra hw help #2

  1. Nov 6, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that {V1--Vn} form an orthonormal basis of R^n for the inner product
    <v,w>= (v^T)Kw for K>0(positive definite) if and only if (A^T)KA=I where A={v1,v2---vn}


    2. Relevant equations
    I don't know what to do in terms of do I write it out using actual matrices or are there some simple properties that I should use like inverse or transposes? Am I missing something?


    3. The attempt at a solution
    (v^T)Kw=0 for K>0 iff (A^T)KA=I <v,w>=(v^T)Kw=0 and ||v||=||w||=1

    [v1--vn][-k1-][v1] = [v1--vn][-k1*v-] = v^2(k1+k2+ --- +kn) = I
    [-k2-][v2] [-k2*v-]
    [ | ][ | ] [ | ]
    [-kn-][vn] [-kn*v-]

    -k1-, -k2-, -kn- are the rows of K
    v1, vn, are the elements of V

    --- represents through (ie. v1--vn means 'V' one through 'V' 'N')

    ||*|| represents the norm(any norm)

    (v^T)Kw represents the quadratic form of the inner product



    I tried doing it by expanding but I don't know where to go from here. I thought about using the inverse rules to get somewhere but I don't think that'll help. Any thoughts are appreciated.
     
  2. jcsd
  3. Nov 6, 2007 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    Let K=[k1 | k2 | ... | kn]. The (i,j)th entry of ATK is viT kj. Now what happens when we multiply ATK by A?
     
  4. Nov 6, 2007 #3
    It becomes (v^T)Kv which when the "subs" i & j for i=j it equals 1 and for i not equal to j it equals 0. Which produces the identity. THANKS


    Matt

    Correct me if I'm wrong.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Linear Algebra hw help #2
  1. Linear Algebra hw help (Replies: 3)

  2. Linear algebra help (Replies: 1)

Loading...