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November 19th, 2015, 02:43 AM   #1
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Optimisation Problem with Rank Constraint

I have a typical least squares problem, i.e I have to find the value of x that minimizes norm of C∗x−d. C is 180x16 matrix, x is 16x1 vector & d is 180x1 vector. However, matrix C is rank deficient (rank(C)=7. Also, I want to represent 16x1 vector x by a 4x4 matrix of rank 1. I have solved it by using normal equation and SVD but the 4x4 matrix that I have obtained from x has rank 4. Is there some constraint that I need to add in my problem in order to get this 4x4matrix of rank 1 from vector x? I would appreciate any suggestion/hint on this type of optimisation problem.
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November 19th, 2015, 08:42 AM   #2
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Linear combinations of the columns of the 4X4 matrix M won't change the rank.
However, it might be possible to find ai to create a matrix polynomial from M of rank 1:

P(M)=a0I+a1M+a2M^2+..

The key question is, how do you find the rank of a matrix polynomial. Nothing on Google.
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