Matrix multiplication to addition

[yiorgos]

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?
Thnx

 

[CompuChip]

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 = 0_{n x m}, Q = A.

 

[yiorgos]

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
http://en.wikipedia.org/wiki/Logarithm_of_a_matrix

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)

 

[HallsofIvy]

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?"

 

[yiorgos]

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?"

So in short you are saying that this transformation doesn't hold?