# Matrix decomposition question

by TimSal
Tags: matrix decomposition
 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 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!
 Sci Advisor HW Helper Thanks P: 26,148 Hi TimSal! Welcome to PF! Hint: multiply both sides by … ?
 P: 5 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.
HW Helper
Thanks
P: 26,148
Matrix decomposition question

Hi TimSal!
So, if (I - A) is invertible, then B = (1 - A)-1AD
 P: 5 Thanks. Any thoughts on the case where A and (I-A) are not invertible?
P: 5
Also, with
 Quote by tiny-tim B = (1 - A)-1AD
there does not seem to be any guarantee that B will indeed be positive semi-definite for any given positive diagonal D.