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Matrix multiplication to addition

  1. Jun 27, 2013 #1
    I am looking for a transformation that relates a matrix product with a matrix addition, e.g.
    AB = PA + QB

    Is there any such transformation?
  2. jcsd
  3. Jun 27, 2013 #2


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    In general, the answer would be no, simply because of a dimensional mismatch.
    Suppose A is an n x m matrix and B is m x r (the first dimension has to be m otherwise AB does not make sense). Then AB is n x r. However, P will be p x n and Q will be q x m, for some numbers p and s (the second coordinate is fixed because the products will have to make sense). All this only works out if p = q = m = r which restricts the validity of the theorem, if it were true, quite a lot.

    Oh, and if you do get the dimensions to work out, there is of course the trivial solution P = 0n x m, Q = A.
  4. Jun 27, 2013 #3
    Thanks for the answer.
    You are right about the particular transformation yet that was just a naive example I gave.

    I am sure there exist "clever" ways to achieve this.

    For example

    I don't know if that helps more but what I actually need is
    the below transformation.
    AB*u -> f(A)*u + f(B)*u

    where u is a vector and f,g() some functions (like the one in the link I gave)
  5. Jun 27, 2013 #4


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    Well, a single number can be thought of as a "one by one" matrix so the first thing you should think about is "if A and B are numbers, do there necessarily exist a function f such that ABu= f(A)u+ f(B)u for every number u?"
  6. Jul 1, 2013 #5
    So in short you are saying that this transformation doesn't hold?
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