e-Ansatz for a Differential Equation in Matrix form

Linear differential equation

$${\bar{q}}^{\prime }(t)=A\bar{q}(t)$$

e-ansatz

$$\boxed{\bar{q}(t)=e^{\lambda t}\bar{v}}$$

where $q(t)$ is a vector-valued function of time, $\lambda$ is a scalar (possibly complex), and $v$ is a constant vector.

The ansatz $q(t)$ is a solution is a solution if $\lambda$ is an eigenvalue of $A$ and $v$ is the corresponding eigenvector.

Note: By definition $Av=\lambda v$

For a diagonizable matrix $A$,

$$A=V\Lambda V^{-1}$$

Where,

• $\Lambda$ is a diagonal matrix with eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ on the diagonal.
• $V$ matrix whose columns are the corresponding eigenvectors of $A$.