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Matrix decomposition question

by TimSal
Tags: matrix decomposition
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TimSal
#1
Jan29-10, 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 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!
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tiny-tim
#2
Jan29-10, 03:22 PM
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Hi TimSal! Welcome to PF!

Hint: multiply both sides by ?
TimSal
#3
Jan30-10, 03:32 AM
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.

tiny-tim
#4
Jan30-10, 04:25 AM
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Matrix decomposition question

Hi TimSal!
Quote Quote by TimSal View Post
AD=(I-A)B
So, if (I - A) is invertible, then B = (1 - A)-1AD
TimSal
#5
Jan30-10, 08:17 AM
P: 5
Thanks. Any thoughts on the case where A and (I-A) are not invertible?
TimSal
#6
Jan30-10, 08:22 AM
P: 5
Also, with
Quote Quote by tiny-tim View Post
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.
tiny-tim
#7
Jan30-10, 08:30 AM
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Quote Quote by TimSal View Post
Thanks. Any thoughts on the case where A and (I-A) are not invertible?
Nope!
Quote Quote by TimSal View Post
Also, with

there does not seem to be any guarantee that B will indeed be positive semi-definite for any given positive diagonal D.
But D isn't given.
TimSal
#8
Jan30-10, 08:40 AM
P: 5
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.


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