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quasi
Posts:
11,708
Registered:
7/15/05


Re: Uniqueness of Q
Posted:
Nov 7, 2012 1:20 PM


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 nonnegative 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



