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Matrix decomposition questionby TimSal
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#1
Jan2910, 02:32 PM

P: 5

Hi everyone,
I have a problem with the following matrix equation: A = B*inv(B+D) where A is a square matrix, B a positive semidefinite matrix, D a positive diagonal matrix and inv() denotes the inverse matrix. All are realvalued. Does anyone know of any simple way to check whether this equation has a solution for given A? And how to obtain this solution? (i.e. find B and D) And whether the solution is unique? Thanks in advance! 


#2
Jan2910, 03:22 PM

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P: 26,160

Hi TimSal! Welcome to PF!
Hint: multiply both sides by … ? 


#3
Jan3010, 03:32 AM

P: 5

Yes, that gives
A(B+D)=B or AD=(IA)B I don't see how that really helps to answer the question though. It's a system of linear equations but I still don't see an easy way of checking whether there exists a solution, nor do I know how to solve this efficiently under the restriction that B is positive semidefinite and D is positive diagonal. 


#4
Jan3010, 04:25 AM

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P: 26,160

Matrix decomposition question
Hi TimSal!



#5
Jan3010, 08:17 AM

P: 5

Thanks. Any thoughts on the case where A and (IA) are not invertible?



#6
Jan3010, 08:22 AM

P: 5

Also, with



#8
Jan3010, 08:40 AM

P: 5

D isn't given, but because there is no guarantee that B will be positive semidefinite for any chosen D, this expression does not help me solve the equation. I still don't know how to pick D and B.



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