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The exponential of a square matrix.

An application field of what we see in this subsection is the solution of certain systems of linear differential equations.

Definition 11.8.1   Let A be a square matrix. The exponential of the matrix A, if it exists, is the matrix

\begin{displaymath}e^{At}= \underset{k=0}{\overset{+ \infty}{\sum}}\frac {1}{k!} A^k
\end{displaymath}

Example 11.8.2   Let $A= \begin{pmatrix}1 & 0 \\ 0 & \frac 12 \end{pmatrix}$. Then:

\begin{displaymath}\forall n \in \mathbb{N} , A^n = \begin{pmatrix}1 & 0 \\ 0 & \frac {1}{2^n} \end{pmatrix}\end{displaymath}

and

\begin{displaymath}\underset{k=0}{\overset{+ \infty}{\sum}}\frac {1}{k!} \begin{...
...matrix}= \begin{pmatrix}1 & 0 \\ 0 & e^{\frac 12} \end{pmatrix}\end{displaymath}

Example 11.8.3   Let $A= \begin{pmatrix}\frac 13 & \frac 16 \\ -\frac 13 & \frac 56 \end{pmatrix}$. This matrix has two distinct eigenvalues, $\frac 12$ and $\frac 23$. The change of basis matrix is $P=\begin{pmatrix}1 & 1 \\ 1 & 2 \end{pmatrix}$. We have:

\begin{displaymath}D=P^{-1}AP = \begin{pmatrix}2 & -1 \\ -1 & 1 \end{pmatrix} \b...
...ix}=\begin{pmatrix}\frac 12 & 0 \\ 0 & \frac 23 \end{pmatrix}.
\end{displaymath}

Thus, we have:

\begin{displaymath}e^{At}=P e^{Dt} P^{-1}=\begin{pmatrix}1 & 1 \\ 1 & 2 \end{pma...
...12}-2e^{\frac 23} & 2e^{\frac 23} - e^{\frac 12}
\end{pmatrix}\end{displaymath}



Noah Dana-Picard
2001-02-26