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Pseudo inverse matrix

  1. Jul 25, 2012 #1
    Hi everybody

    I have a question that I have a guess for the answer but I want to be sure

    I have an identity

    C=A*B

    where A,B,C are matrices and in general they are not square

    is there a way to find A in terms of B and C by using pseudo inverse matrix or using pseudo inverse might help me in any ways for finding A.

    my guess is: 'No, pseudo matrix inverse will not help in anyway'

    If pseudo inverse will not work I am planning to do a gradient search for finding the entries of A (do you have any better option?)

    If possible say some words about the uniqueness problem. for given A and B, C is unique, but for given B and C, is A unique

    Thanks for the help
     
    Last edited: Jul 25, 2012
  2. jcsd
  3. Jul 26, 2012 #2
    Hi

    I found solution to my problem, let me share it in case someone else might need it;

    lets assume C=A*B
    and A, B, C are matrices with dimensions $$ A_{(m,n)}, B_{(n,k)}, C_{(m,k)} $$

    Assume we are trying to find A for given B and C

    Then the number of equations we have is m*k, one equation for each entry of C matrix
    The number of unknowns are m*n, each entry in the A matrix is an unknown

    So if number of equations is greater then or equal to the number of unknowns then there is a unique solution. In other words if [itex]m*k \geq m*n[/itex] or simply [itex]k \geq n[/itex] then there is a unique solution.

    If there is a unique solution then

    C=A*B can be re written as A=C*pinv(B)

    where pinv stands for 'pseudo inverse'

    if there is no unique solution; i.e [itex]k<n[/itex], then one need to go to gradient search and statistical methods to find possible candidates for A matrix
     
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