Matrix exponential

For a square matrix $A$, its exponential is the convergent power seriesIt is the fundamental matrix for $y^{\prime }=Ay$ and satisfies $e^{(s+t)A}=e^{sA}{e}^{tA}$.

The determinant is a smooth Lie group homomorphism from the real general linear group to ${\mathbb{R}}^{\times }$. Its derivative at the identity is the trace, and the flow-defined exponential on ${\mathbb{R}}^{\times }$ is the ordinary scalar exponential. Naturality of the Lie group exponential therefore proves the identity without diagonalizability assumptions. The series ${\sum }_{k\ge 0}{t}^k{A}^k/k!$ is the flow-defined matrix exponential because it solves $E^{\prime }=EA$, $E(0)=I$, and has inverse $E(-t)$.

If a real square matrix satisfies $M^2=-I$, split the absolutely convergent matrix exponential into even and odd powers to obtain $e^{tM}=I\cos t+M\sin t$. It is the fundamental matrix for ${\mathbf{x}}^{\prime }=M\mathbf{x}$. Consequently every solution has period $2\pi$, and the eigenvalues over the complex numbers belong to $\{i,-i\}$.

The Baker--Campbell--Hausdorff formula expresses the local matrix logarithm of a product of exponentials as

On its half-plane of convergence,Indeed, applying the transform to $y^{\prime }=Ay$, $y(0)=y_0$, gives $(sI-A)Y=y_0$ for every $y_0$.