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Kaba
Posts:
289
Registered:
5/23/11
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Re: Uniqueness of Q
Posted:
Nov 7, 2012 12:43 PM
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7.11.2012 18:01, Rupert wrote:
> On Nov 7, 4:39 pm, Kaba <k...@nowhere.com> wrote:
>> 7.11.2012 17:30, Kaba wrote:
>>> Hi,
>>
>>> Let Q, U, D, V in R^{n x n}, where Q^T Q = U^T U = V^T V = I, and D is
>>> non-negative diagonal. Consider the equation
>>
>>> Q^T UDV^T = VDU^T Q.
>>
>>> One solution to this equation is Q = UV^T.
>>
>>> Prove or disprove: this solution is unique.
>>
>> Disproved: Q = -UV^T is also a solution.
>>
>> Prove or disprove: UV^T and -UV^T are the only solutions.
>>
>> --http://kaba.hilvi.org
>
> Would it not be the case that any scalar multiple of UV^T is a
> solution?
>
> Also I think UD'V^T is a solution whenever D' is diagonal.
Oops, take that back. The solution Q = UD'V^T needs to orthogonal, so
all the elements in the diagonal of D' must be either +1 or -1. Anyway,
are _these_ all solutions?
--
http://kaba.hilvi.org
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