Flow and convergence

[tommy1729]

the matrix multiplication has issues if you mean carleman and such.

This follows from the many sqrt a matrix can have.

if the eigenvalues are nonreal the n th root of the matrix can have n solutions PER eigenvalue.

So the n th root of a m*m matrix can have up to 

n^m

SO 

a matrix of size 10 can have

2^10 sqrt roots.

and that does not consider the one-periodic function.

So unless you define a way to get the correct roots ( such as closest to the + reals )
and that works out nice , even with increasing matrix size

then and only then  may you have a good definition.



It has been suggested to use 

A^(1/n) = exp( 1/n ln(A) )



this way maybe you have the semi group iso , IF the ln and exp do not give issues.
( again by conjugate composition )

But this seems only consistant when locally around a fixpoint :

so basically equivalent to the fixpoint expansion like koenings function ( assuming f ' (fix ) is ok )



regards

tommy1729