Eikonal Blog


Magnus’ Bracket Operator

Filed under: mathematics — Tags: — sandokan65 @ 22:46

Definition: Define z(x,y) via product of exponentials e^z :\equiv e^x e^y.


  • z = x+y+\frac12 [x,y] + \frac1{12} (\{xy^2\} + \{yx^2\}) + \frac1{24}\{xy^2x\} + \cdots
  • e^{-y} e^x e^y = e^{x+\frac1{1!}[x,y] + \frac1{24}\{xy^2\}+\cdots} = e^{\{x e^y\}_M}

Definition: Here “the bracket operator” \{\cdots\}_M is defined by following properties:

  1. \{G\}_M = G if G is homogeneous of the first degree,
  2. \{r G\}_M = r \{G\}_M if r is a rational number,
  3. \{G_1 + G_2\}_M = \{G_1\}_M + \{G_2\}_M,
  4. \{G x\}_M = [\{G\}_M , x], \{G y\}_M = [\{G\}_M , y]


  • \{x^2 G\}_M = \{y^2 G\}_M = 0,
  • \{x y^n\}_M = x y^n - \binom{n}{1} y x y^{n-1} + \cdots + (-)^k \binom{n}{k}^k y^k x y^{n-k} + \cdots + (-)^n y^n x.
  • \{\omega_n\}_M = n \omega_n for \omega_n homogeneous in x and y of degree n, i.e. \{\omega\}_M = \left(x D_x + y D_y\right) \omega.

Source: T1272 = W. Magnus “A Connection between the Baker-Hausdorff formula and a Problem of Burnside”; Annals of Mathematics; Vol 52, No 1, July 1950.

Related here: Magnus on matrix exponentials – https://eikonal.wordpress.com/2010/01/26/magnus-on-matrix-exponentials/


Magnus on matrix exponentials

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


  • 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

    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



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/

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