Functions of Finite Polynomials

Motivation: We want to efficiently compute $e^{At}$.

1. An infinite series is inconvenient $e^{At}={\sum }_{k=0}^{\infty }\frac{t^k}{k!}{A}^k$
2. Using the Cayley Hamilton Theorem provides the recipe to use a finite series

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For example, let’s say you want to compute

$$A=\left[\begin{array}{cc}2& 4\\ 0& 1\end{array}\right],f(A)=e^{At}$$

1. Compute eigenvalues
   • ${\lambda }_1=2,{\lambda }_2=1$
2. Solve the set of equations $h(\lambda )={\beta }_0+{\beta }_1\lambda$, $f(\lambda )=e^{\lambda t}$
   • ${\beta }_0=-e^t(-2+e^t)$, and ${\beta }_1=e^t(-1+e^t)$
3. Solve $f(A)={\beta }_0\mathbb{I}+{\beta }_{1}A$
   • $f(A)=-e^t(-2+e^t)\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+e^t(-1+e^t)\left[\begin{array}{cc}2& 4\\ 0& 1\end{array}\right]=e^{At}$