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 semi-definite matrix, D a positive diagonal matrix and inv() denotes the inverse matrix. All are real-valued. 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!
Yes, that gives A(B+D)=B or AD=(I-A)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 semi-definite and D is positive diagonal.
Also, with there does not seem to be any guarantee that B will indeed be positive semi-definite for any given positive diagonal D.
D isn't given, but because there is no guarantee that B will be positive semi-definite for any chosen D, this expression does not help me solve the equation. I still don't know how to pick D and B.