Consider a system of [1st order ODEs](1st%20order%20ODE) where we want to solve for vector $\mathbf{v}(t)$. The differential equation is $\frac{d\mathbf{v}(t)}{dt} = X\mathbf{v}(t)$ The [matrix exponential](Matrix%20exponentials.md) is the solution Where with $\mathbf{v}(0) = \mathbf{v}_0$ the solution is _unique_ and takes the form $\mathbf{v}(t) = e^{tX}$. # Solutions in $\mathbb{C}$ [Complex matrix exponentials](Complex%20matrix%20exponentials.md) --- # Proofs and examples ## Proof of the existence of unique solutions to 1st order matrix ODEs ## Example of a matrix ODE in $\mathbb{R}$ We can trace out the flow of the vector field for a couple of example forms of $X$ $X_1 = \begin{pmatrix} 1 & 2 \\ -2 & 1 \\ \end{pmatrix}\;\;\;\;\;\;X_2 = \begin{pmatrix} 1 & 2 \\ 2 & 1 \\ \end{pmatrix}$ With the following resulting vector field curves:  #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices #MathematicalFoundations/Analysis/DifferentialEquations/OrdinaryDifferentialEquations