Example 1.  Consider the matrix  ,
1 (a)  Find  .
1 (b)  Find  .

Solution 1 (b).

We want to find

First look at some powers

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Now use the calculation

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Find the expression for the general term

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Find matrix exponential     will be the sum of the infinite series

The sum of the first five terms is

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The sum of the first ten terms is

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Each element in     can be calculated by the sum of an infinite series and Mathematica can assist us in these computations.

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Therefore, the matrix exponential     is

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This can be compared to the matrix exponential    that can be computed by using Mathematica's built in procedure  MatrixExp[At].

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Caveat.  This shows the power of "artificial intelligence" that is available in Mathematica.  For this example the matrix had a full set of eigenvectors.  Mathematica is "smart" enough to know how to compute the matrix exponential for the difficult case when the set of eigenvectors is deficient.

(c) John H. Mathews 2004