Definition: The Magnus’ bracket is defined as .
- for a power series one has
where the Hausdorff polarization operator is defined as following:
- Lemma: For two power series and s/t one has the following equivalence:
For one has:
The Magnus’ formula for ordered exponentials
And this is really beautiful thing: for the ordered exponential one can observe its logarithm (the Magnus function of A) (i.e. ) and get following nonlinear ODE for it:
where (), and ‘s are ordinary Bernoulli’s numbers.
Source: T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematics, Vol VII, 6490673 (1954). Note: This paper is probably mother of the whole this area of matrix exponentials.
Related here: Magnus’ Bracket Operator – https://eikonal.wordpress.com/2010/02/15/magnus-bracket-operator/