---
tag: EBV9M9C435
kind: concept
name: Magnus expansion
msc: [34A25, 34A30]
sources:
  - url: https://en.wikipedia.org/wiki/Magnus_expansion
same_as:
  wikidata: Q6732207
  wikipedia: Magnus expansion
---

## Summary

The Magnus expansion writes the solution of a linear system $Y' = A(t)\, Y$, $Y(0) = I$, as one [[matrix exponential|020M04FN1H]] $Y(t) = e^{\Omega(t)}$.
The exponent $\Omega(t)$ is a series of iterated integrals of commutators of $A$ at different times.
When the matrices $A(t)$ commute with each other, only the first term $\int_0^t A$ remains.
Truncations of the series keep structure, for example they stay in a matrix group, which makes them useful in numerical methods and quantum mechanics.
See the [[definition|TD9T02HKPD]].

## Explanations

### L1 human

For a scalar equation $y' = a(t)\, y$ the solution is $y = e^{\int a}$.
For matrices this formula is wrong in general, because matrices at different times need not commute.
The Magnus expansion corrects the exponent by extra terms built from commutators $[A(t_1), A(t_2)] = A(t_1) A(t_2) - A(t_2) A(t_1)$.
These terms measure how much the matrices fail to commute.