Eikonal Blog

2010.01.26

Magnus on matrix exponentials

Definition: The Magnus’ bracket is defined as $\{y,x^n\}_M :\equiv (- ad(x))^n y$.

Properties:

• $e^{-ad(x)} y = \sum_{n=0}^\infty \{y, x^n\}_M$
• for a power series $P(x)=\sum_{n=0}^\infty p_l x^l$ one has $\{y, P(x)\}_M = \sum_{n=0}^\infty p_l \{y, x^n\}_M$
• Hausdorf:
• $e^{-x} (y D_x) e^x = \left\{y,\frac{e^x -1}{x}\right\}_M$,
• $(y D_x e^{-x}) e^x = \left\{y,\frac{1-e^{-x}}{x}\right\}_M$,

where the Hausdorff polarization operator $(y D_x)$ is defined as following:

$(y D_x) x^n :\equiv \sum_{k=0}^{n-1} x^k y x^{n-k-1}$.

• Lemma: For two power series $P(x)$ and $Q(x)$ s/t $P(x) Q(x) = 1$ one has the following equivalence:
$\{y,P(x)\}_M=u \Leftrightarrow y = \{u, Q(x)\}_M$.
• $\{\{y,x^n\}_M,x^m\}_M = \{y,x^{n+m}\}_M$.

For $e^{z} :\equiv e^{x}e^{y}$ one has:

• $z=\ln(1+u) = u - \frac12 u^2 + \frac13 u^3 + \cdots$

where
$u:\equiv e^x e^y - 1 = x + y + \frac12 x^2 + xy + \frac12 y^2 + \cdots$
• $z = x + y + \frac12 [x,y] + \frac1{12} \{x,y^2\}_M + \frac1{12} \{y,x^2\}_M -\frac1{24} [\{x,y^2\}_M,x] - \frac1{720} \{x,y^4\}_M - \frac1{720} \{y,x^4\}_M + \frac1{180} [[\Delta_1,x],y] - \frac1{180}\{\Delta_1,y^2\}_M - \frac1{120} [\Delta_1,\Delta] - \frac1{360} [\Delta_2,\Delta] + \cdots$

where

$\Delta=[x,y]$,
$\Delta_1=[\Delta,x]$,
$\Delta_2=[\Delta,y]$.

The Magnus’ formula for ordered exponentials

And this is really beautiful thing: for the ordered exponential $U(t|A):\equiv \left(e^{\int_0^t dt_1 A(t_1)}\right)_+$ one can observe its logarithm (the Magnus function of A) $\Omega(t|A) :\equiv \ln U(t|A)$ (i.e. $e^{\Omega}=U$) and get following nonlinear ODE for it:

$\frac{d\Omega(t)}{dt} = \{A, \frac{\Omega}{1-e^{-\Omega}}\}_M = \sum_{n=0}^\infty \beta_n \{A,\Omega^n\}_M = A + \frac12 [A,\Omega] + \frac1{12}\{A,\Omega^2\}_M + \cdots$

where $\beta_{2n1+1}=0$ ($\forall n\in{\Bbb N}_0$), $\beta_{2n} = (-)^{n-1} \frac{B_{2n}}{(2n)!}$ and $B_n$‘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/