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Homework Help: 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


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    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


    Correct me if I'm wrong.
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