Derivative of Matrix Exponential

Properties

For a square matrix $A$, the derivative of the matrix exponential $e^{At}$ with respect to $t$ is:

$\frac{d}{dt}{e}^{At}=A{e}^{At}=e^{At}A$

Equivalently:

$\frac{\partial }{\partial t}{e}^{At}=e^{At}A$

Interpretation

This property mirrors the scalar case $\frac{d}{dt}{e}^{at}=a{e}^{at}$. The matrix $A$ can be placed on either the left or right because $A$ commutes with its own exponential: $A{e}^{At}=e^{At}A$.

Role in Kolmogorov Equations

This property is used to verify that $P(t)=e^{Qt}$ solves the Kolmogorov differential equations:

• Backward equation: $\frac{d}{dt}P(t)=Q{e}^{Qt}=QP(t)$ ✓
• Forward equation: $\frac{d}{dt}P(t)=e^{Qt}Q=P(t)Q$ ✓

Also, $P(0)=e^{Q\cdot 0}=I$, satisfying the initial condition.

Related

• Matrix Exponential
• Kolmogorov Differential Equations