Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[MatrixExponential], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The matrix exponential, ${\ⅇ}^{At}$, is a matrix with the same shape as A and is defined as follows: ${\ⅇ}^{At}=I+At+\frac{1}{2}\!A^2{t}^2+...$ where I is the identity matrix.

If the second parameter is not given, then the first indeterminate (if any) in the matrix is removed and used as a parameter.

The exponential function can only return a symbolic answer if the eigenvalues of A can be found. To get a floating-point approximation, use at least one floating-point entry in A.

The command with(linalg,exponential) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$A≔\mathrm{array}\left(\left[\left[t\,0\,0\right]\,\left[0\,t\,0\right]\,\left[0\,0\,t\right]\right]\right)$

$A≔\left[\begin{array}{ccc}t& 0& 0\\ 0& t& 0\\ 0& 0& t\end{array}\right]$

$\mathrm{exponential}\left(A\right)$

$\left[\begin{array}{ccc}{\ⅇ}^t& 0& 0\\ 0& {\ⅇ}^t& 0\\ 0& 0& {\ⅇ}^t\end{array}\right]$

$B≔\mathrm{array}\left(\left[\left[-13\,-10\right]\,\left[21\,16\right]\right]\right)$

$B≔\left[\begin{array}{cc}\mathrm{-13}& \mathrm{-10}\\ 21& 16\end{array}\right]$

$\mathrm{exponential}\left(B\,t\right)$

$\left[\begin{array}{cc}15{\ⅇ}^t-14{\ⅇ}^{2t}& -10{\ⅇ}^{2t}+10{\ⅇ}^t\\ 21{\ⅇ}^{2t}-21{\ⅇ}^t& -14{\ⅇ}^t+15{\ⅇ}^{2t}\end{array}\right]$