Expediting the Calculation of Exponential Matrix

Where D is the diagonal matrix of order NxN with the N distinct eigenvalues (Xk, i = 1..N) as its diagonal elements. P is the eigenvector matrix defined as 

5 Method of Lines for Parabolic Partial Differential Equations 

Since D is a diagonal matrix, the exponential matrix of D is easily obtained as  

Maple can be used to obtain the eigenvalues and eigenvector matrix (P). Maple takes only a few seconds to calculate the eigenvalues (for a 20 x 20 matrix it takes less than a second). However, Maple takes a long time to calculate the eigenvector matrix. To overcome this problem, we can obtain the particular eigenvector Xk using the equation 

The following procedure in Maple can be used to obtain exponential matrix for any matrix with distinct eigenvalues. 

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In section 5.1.7, a procedure to expedite the calculation of exponential matrix was developed. This procedure is valid as long as all the eigenvalues are distinct. 

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