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Topic: Uniqueness of Q
Replies: 5   Last Post: Nov 7, 2012 1:20 PM

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quasi

Posts: 10,226
Registered: 7/15/05
Re: Uniqueness of Q
Posted: Nov 7, 2012 1:20 PM
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Kaba wrote:
>Kaba wrote:
>>
>> 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.


Let's test your claim using the simplest possible choices for
U,V,D subject to the requirements

U^T U = V^T V = I

D nonnegative diagonal

Thus, let U,V,D all be equal to I (the nxn identity matrix).

Then the equation

Q^T UDV^T = VDU^T Q

reduces to just

Q^T = Q

and consequently, the requirement

Q^T Q = I

reduces to just Q^2 = I.

The conditions

Q^T = Q

Q^2 = I

are satisfied for Q = I and Q = -I but there are other
solutions as well. For example, Q can be any diagonal matrix
with diagonal entries equal to plus or minus 1 in any
combination, thus disproving your claim.

Other counterexamples can be found as well.

For example, Q could be any permutation matrix of order 2,
that is, Q could be a transposition matrix or a product of
disjoint transposition matrices.

quasi



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