- #1
blue2004STi
- 21
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Homework Statement
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}
Homework 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?
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.