Matrix Exponential

Matrix Exponential

The matrix exponential $e^A$ extends the scalar exponential to matrices via the Taylor series:

$e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \frac{A^4}{4!} + \cdots$

Applications

• Differential equations: the solution to $\dot{x} = Ax$ with $x(0) = x_0$ is $x(t) = e^{tA} x_0$
• Control theory: state transition matrices
• Quantum mechanics: time evolution operator

Algorithm

Accumulate terms until convergence:

$\text{result} = I, \quad \text{term} = I, \quad \text{for } k = 1, 2, 3, \ldots: \quad \text{term} = \text{term} \times A / k, \quad \text{result} += \text{term}$

Example: Rotation Generator

The matrix $A = \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$ is the generator of 2D rotation. Its exponential is exactly the rotation matrix:

$e^A = \begin{pmatrix}\cos(1) & \sin(1) \\ -\sin(1) & \cos(1)\end{pmatrix} \approx \begin{pmatrix}0.5403 & 0.8415 \\ -0.8415 & 0.5403\end{pmatrix}$

Your Task

Implement mat_exp(A, terms=20) using the Taylor series. Use matrix multiplication to compute successive powers of A.