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quasi
Posts:
9,093
Registered:
7/15/05
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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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