Matrix exponential

Consider a matrix $A \in \R^{n \times n}$ . From the Taylor series definition of the exponential we can write

e^A = \sum_{k=0}^\infty \frac{A^k}{k!}.

If $A$ can be diagonalized we have

A^k = V \Lambda V^{-1} V \Lambda V^{-1} \cdots V \Lambda V^{-1} = V \Lambda^k V^{-1}

since $V^{-1} V = I$ . It follows that

e^A = \sum_{k=0}^\infty V \frac{\Lambda^k}{k!} V^{-1} = V \left(\sum_{k=0}^\infty \frac{\Lambda^k}{k!} \right) V^{-1} = V e^\Lambda V^{-1}.

Since $\Lambda$ is a diagonal matrix, raising it to some power is equivalent to raising every entry along the diagonal to that power.

\Lambda^k = \mathrm{diag}(\lambda_1^k, \lambda_2^k, \ldots, \lambda_n^k).

Fundamental matrix

The method we saw for solving linear ODEs with constant coefficients actually generalizes to linear ODE systems. Instead of a scalar solution $x=e^{\lambda t}c$ , for a system of ODEs described by a constant matrix $A$ , we write the solution $\mathbf x = e^{At} \mathbf C$ . Here $e^{At}$ is called the fundamental matrix.

The columns of the fundamental matrix are the solutions to the ODE system.