Math 2.0

Ordinary differential equations Theorem

Matrix exponential as the solution of a linear system

Builds on Homogeneous linear system with constant coefficientsInitial value problem

Statement

Let AA be a real n×nn \times n matrix, and for t∈ℝt \in \mathbb{R} let etA=∑k=0∞tkk!Ake^{t A} = \sum_{k=0}^{\infty} \frac{t^k}{k!} A^k, a series that converges for each tt. Then the map t↦etAt \mapsto e^{t A} is differentiable on ℝ\mathbb{R}, and ddt etA=A etA=etAA.\frac{d}{dt}\, e^{t A} = A\, e^{t A} = e^{t A} A. Consequently, for each x0∈ℝnx_0 \in \mathbb{R}^n, the function x(t)=etAx0x(t) = e^{t A} x_0 is the unique solution on ℝ\mathbb{R} of the initial value problem x′=Axx' = A x, x(0)=x0x(0) = x_0.

Remarks

Uniqueness: if xx is a solution, then ddt(e−tAx(t))=0\frac{d}{dt}\left(e^{-tA} x(t)\right) = 0, so e−tAx(t)=x0e^{-tA} x(t) = x_0. If A=PDP−1A = P D P^{-1} with DD diagonal, then etA=PetDP−1e^{tA} = P e^{tD} P^{-1}, and etDe^{tD} is diagonal with the entries etλke^{t \lambda_k}. The matrix etAe^{tA} is the fundamental matrix of x′=Axx' = A x with value II at t=0t = 0.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionSystem of linear differential equations, definition, no evidence levelSystem of linear diff…definitionHomogeneous linear system with constant coefficients, definition, no evidence levelHomogeneous linear sy…definitionMatrix exponential as the solution of a linear system, statement, no evidence levelMatrix exponential as…statement
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Order of a differential equation definition
  6. Solution of an ordinary differential equation definition
  7. Initial value problem definition
  8. Linear differential equation of order n definition
  9. System of linear differential equations definition
  10. Homogeneous linear system with constant coefficients definition
  11. Matrix exponential as the solution of a linear system statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_exponential, Linear differential equation systems
About this page
Tag0YC8JT513J
URIm2://0YC8JT513J
Kindstatement, theorem
WritingW0 stub
MSC34A30, 15A16
StatusThis copy of the site has no build data. Run m2 status 0YC8JT513J.
UsesHomogeneous linear system with constant coefficients, Initial value problem

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